Misplaced Pages

#533466

94-462: The characteristic polynomial of an n  ×  n matrix A is defined as p A ( λ ) = det ( λ I n − A ) {\displaystyle p_{A}(\lambda )=\det(\lambda I_{n}-A)} , where det is the determinant operation, λ is a variable scalar element of the base ring , and I n is the n  ×  n identity matrix . Since each entry of

188-427: A n × n {\displaystyle n\times n} matrix is monic (its leading coefficient is 1 {\displaystyle 1} ) and its degree is n . {\displaystyle n.} The most important fact about the characteristic polynomial was already mentioned in the motivational paragraph: the eigenvalues of A {\displaystyle A} are precisely

282-427: A 1 , 1 − a 1 , 2 ⋯ − a 1 , n − a 2 , 1 t − a 2 , 2 ⋯ − a 2 , n ⋮ ⋮ ⋱ ⋮ − a n , 1 −

376-424: A k x k {\displaystyle f(x)=\sum _{k=0}^{\infty }a_{k}x^{k}} and the characteristic polynomial p ( x ) of degree n of an n  ×  n matrix A , the function can be expressed using long division as f ( x ) = q ( x ) p ( x ) + r ( x ) , {\displaystyle f(x)=q(x)p(x)+r(x),} where q ( x )

470-724: A n , 2 ⋯ t − a n , n | = t n + c n − 1 t n − 1 + ⋯ + c 1 t + c 0 , {\displaystyle {\begin{aligned}p(t)&=\det(tI_{n}-A)={\begin{vmatrix}t-a_{1,1}&-a_{1,2}&\cdots &-a_{1,n}\\-a_{2,1}&t-a_{2,2}&\cdots &-a_{2,n}\\\vdots &\vdots &\ddots &\vdots \\-a_{n,1}&-a_{n,2}&\cdots &t-a_{n,n}\end{vmatrix}}\\[5pt]&=t^{n}+c_{n-1}t^{n-1}+\cdots +c_{1}t+c_{0},\end{aligned}}} Characteristic polynomial In linear algebra ,

564-802: A + d ) ) + ( a d − b c ) I 2 = ( b c − a d 0 0 b c − a d ) + ( a d − b c ) I 2 = ( 0 0 0 0 ) {\displaystyle {\begin{aligned}&{}A^{2}-(a+d)A+(ad-bc)I_{2}\\[1ex]&={\begin{pmatrix}a^{2}+bc&ab+bd\\ac+cd&bc+d^{2}\\\end{pmatrix}}-{\begin{pmatrix}a(a+d)&b(a+d)\\c(a+d)&d(a+d)\end{pmatrix}}+(ad-bc)I_{2}\\[1ex]&={\begin{pmatrix}bc-ad&0\\0&bc-ad\\\end{pmatrix}}+(ad-bc)I_{2}\\[1ex]&={\begin{pmatrix}0&0\\0&0\end{pmatrix}}\end{aligned}}} For

658-412: A + d ) A + ( a d − b c ) I 2 = ( a 2 + b c a b + b d a c + c d b c + d 2 ) − ( a ( a + d ) b ( a + d ) c ( a + d ) d (

752-411: A + d ) A + ( a d − b c ) I 2 = ( 0 0 0 0 ) ; {\displaystyle p(A)=A^{2}-(a+d)A+(ad-bc)I_{2}={\begin{pmatrix}0&0\\0&0\end{pmatrix}};} which is indeed always the case, evident by working out the entries of A . A 2 − (

846-411: A zero matrix or null matrix is a matrix all of whose entries are zero . It also serves as the additive identity of the additive group of m × n {\displaystyle m\times n} matrices, and is denoted by the symbol O {\displaystyle O} or 0 {\displaystyle 0} followed by subscripts corresponding to the dimension of

940-2352: A ) , the characteristic polynomial is given by p ( λ ) = λ − a , and so p ( A ) = ( a ) − a (1) = 0 is trivial. As a concrete example, let A = ( 1 2 3 4 ) . {\displaystyle A={\begin{pmatrix}1&2\\3&4\end{pmatrix}}.} Its characteristic polynomial is given by p ( λ ) = det ( λ I 2 − A ) = det ( λ − 1 − 2 − 3 λ − 4 ) = ( λ − 1 ) ( λ − 4 ) − ( − 2 ) ( − 3 ) = λ 2 − 5 λ − 2. {\displaystyle {\begin{aligned}p(\lambda )&=\det(\lambda I_{2}-A)=\det \!{\begin{pmatrix}\lambda -1&-2\\-3&\lambda -4\end{pmatrix}}\\&=(\lambda -1)(\lambda -4)-(-2)(-3)=\lambda ^{2}-5\lambda -2.\end{aligned}}} The Cayley–Hamilton theorem claims that, if we define p ( X ) = X 2 − 5 X − 2 I 2 , {\displaystyle p(X)=X^{2}-5X-2I_{2},} then p ( A ) = A 2 − 5 A − 2 I 2 = ( 0 0 0 0 ) . {\displaystyle p(A)=A^{2}-5A-2I_{2}={\begin{pmatrix}0&0\\0&0\\\end{pmatrix}}.} We can verify by computation that indeed, A 2 − 5 A − 2 I 2 = ( 7 10 15 22 ) − ( 5 10 15 20 ) − ( 2 0 0 2 ) = ( 0 0 0 0 ) . {\displaystyle A^{2}-5A-2I_{2}={\begin{pmatrix}7&10\\15&22\\\end{pmatrix}}-{\begin{pmatrix}5&10\\15&20\\\end{pmatrix}}-{\begin{pmatrix}2&0\\0&2\\\end{pmatrix}}={\begin{pmatrix}0&0\\0&0\\\end{pmatrix}}.} For

1034-457: A fundamental role, since, given a linear transformation , an eigenvector is a vector whose direction is not changed by the transformation, and the corresponding eigenvalue is the measure of the resulting change of magnitude of the vector. More precisely, suppose the transformation is represented by a square matrix A . {\displaystyle A.} Then an eigenvector v {\displaystyle \mathbf {v} } and

SECTION 10

#1732792980534

1128-415: A further example, when considering f ( A ) = e A t w h e r e A = ( 0 1 − 1 0 ) , {\displaystyle f(A)=e^{At}\qquad \mathrm {where} \qquad A={\begin{pmatrix}0&1\\-1&0\end{pmatrix}},} then the characteristic polynomial is p ( x ) = x  + 1 , and

1222-722: A general n  ×  n invertible matrix A , i.e., one with nonzero determinant, A can thus be written as an ( n − 1) -th order polynomial expression in A : As indicated, the Cayley–Hamilton theorem amounts to the identity p ( A ) = A n + c n − 1 A n − 1 + ⋯ + c 1 A + ( − 1 ) n det ( A ) I n = 0. {\displaystyle p(A)=A^{n}+c_{n-1}A^{n-1}+\cdots +c_{1}A+(-1)^{n}\det(A)I_{n}=0.} The coefficients c i are given by

1316-574: A general n  ×  n matrix, provided no root be zero, relies on the following alternative expression for the determinant , p ( λ ) = det ( λ I n − A ) = λ n exp ⁡ ( tr ⁡ ( log ⁡ ( I n − A / λ ) ) ) . {\displaystyle p(\lambda )=\det(\lambda I_{n}-A)=\lambda ^{n}\exp(\operatorname {tr} (\log(I_{n}-A/\lambda ))).} Hence, by virtue of

1410-414: A generic 2 × 2 matrix, A = ( a b c d ) , {\displaystyle A={\begin{pmatrix}a&b\\c&d\\\end{pmatrix}},} the characteristic polynomial is given by p ( λ ) = λ − ( a + d ) λ + ( ad − bc ) , so the Cayley–Hamilton theorem states that p ( A ) = A 2 − (

1504-427: A polynomial f ( t ) = t 3 + 1 , {\displaystyle f(t)=t^{3}+1,} for example, is evaluated on a matrix A {\displaystyle A} simply as f ( A ) = A 3 + I . {\displaystyle f(A)=A^{3}+I.} The theorem applies to matrices and polynomials over any field or commutative ring . However,

1598-599: A square n × n {\displaystyle n\times n} matrix and let f ( t ) {\displaystyle f(t)} be a polynomial. If the characteristic polynomial of A {\displaystyle A} has a factorization p A ( t ) = ( t − λ 1 ) ( t − λ 2 ) ⋯ ( t − λ n ) {\displaystyle p_{A}(t)=(t-\lambda _{1})(t-\lambda _{2})\cdots (t-\lambda _{n})} then

1692-447: A system of n linear equations , which can be solved to determine the coefficients c i . Thus, one has f ( A ) = ∑ k = 0 n − 1 c k A k . {\displaystyle f(A)=\sum _{k=0}^{n-1}c_{k}A^{k}.} When the eigenvalues are repeated, that is λ i = λ j for some i ≠ j , two or more equations are identical; and hence

1786-518: A time scale of a century, that is, slow compared to annual motion) of planetary orbits, according to Lagrange 's theory of oscillations. Secular equation may have several meanings. The above definition of the characteristic polynomial of a matrix A ∈ M n ( F ) {\displaystyle A\in M_{n}(F)} with entries in a field F {\displaystyle F} generalizes without any changes to

1880-412: Is det ( − A ) = ( − 1 ) n det ( A ) , {\displaystyle \det(-A)=(-1)^{n}\det(A),} the coefficient of t n {\displaystyle t^{n}} is one, and the coefficient of t n − 1 {\displaystyle t^{n-1}} is tr(− A ) = −tr( A ) , where tr( A )

1974-716: Is m × m {\displaystyle m\times m} and B A {\displaystyle BA} is n × n {\displaystyle n\times n} matrix, and one has p B A ( t ) = t n − m p A B ( t ) . {\displaystyle p_{BA}(t)=t^{n-m}p_{AB}(t).\,} To prove this, one may suppose n > m , {\displaystyle n>m,} by exchanging, if needed, A {\displaystyle A} and B . {\displaystyle B.} Then, by bordering A {\displaystyle A} on

SECTION 20

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2068-721: Is det ( t I − A ) = ( t − cosh ⁡ ( φ ) ) 2 − sinh 2 ⁡ ( φ ) = t 2 − 2 t   cosh ⁡ ( φ ) + 1 = ( t − e φ ) ( t − e − φ ) . {\displaystyle \det(tI-A)=(t-\cosh(\varphi ))^{2}-\sinh ^{2}(\varphi )=t^{2}-2t\ \cosh(\varphi )+1=(t-e^{\varphi })(t-e^{-\varphi }).} The characteristic polynomial p A ( t ) {\displaystyle p_{A}(t)} of

2162-1026: Is p ( x ) = ( x − 1)( x − 3) = x − 4 x + 3 , and the eigenvalues are λ = 1, 3 . Let r ( x ) = c 0 + c 1 x . Evaluating f ( λ ) = r ( λ ) at the eigenvalues, one obtains two linear equations, e = c 0 + c 1 and e = c 0 + 3 c 1 . Solving the equations yields c 0 = (3 e − e )/2 and c 1 = ( e − e )/2 . Thus, it follows that e A t = c 0 I 2 + c 1 A = ( c 0 + c 1 2 c 1 0 c 0 + 3 c 1 ) = ( e t e 3 t − e t 0 e 3 t ) . {\displaystyle e^{At}=c_{0}I_{2}+c_{1}A={\begin{pmatrix}c_{0}+c_{1}&2c_{1}\\0&c_{0}+3c_{1}\end{pmatrix}}={\begin{pmatrix}e^{t}&e^{3t}-e^{t}\\0&e^{3t}\end{pmatrix}}.} If, instead,

2256-537: Is Rodrigues' rotation formula . For the notation, see 3D rotation group#A note on Lie algebras . More recently, expressions have appeared for other groups, like the Lorentz group SO(3, 1) , O(4, 2) and SU(2, 2) , as well as GL( n , R ) . The group O(4, 2) is the conformal group of spacetime , SU(2, 2) its simply connected cover (to be precise, the simply connected cover of the connected component SO(4, 2) of O(4, 2) ). The expressions obtained apply to

2350-445: Is non-singular this result follows from the fact that A B {\displaystyle AB} and B A {\displaystyle BA} are similar : B A = A − 1 ( A B ) A . {\displaystyle BA=A^{-1}(AB)A.} For the case where both A {\displaystyle A} and B {\displaystyle B} are singular,

2444-425: Is similar to f ( U ) , {\displaystyle f(U),} it has the same eigenvalues, with the same algebraic multiplicities. The term secular function has been used for what is now called characteristic polynomial (in some literature the term secular function is still used). The term comes from the fact that the characteristic polynomial was used to calculate secular perturbations (on

2538-1748: Is upper triangular with λ 1 , … , λ n {\displaystyle \lambda _{1},\ldots ,\lambda _{n}} on the diagonal (with each eigenvalue repeated according to its algebraic multiplicity). (The Jordan normal form has stronger properties, but these are sufficient; alternatively the Schur decomposition can be used, which is less popular but somewhat easier to prove). Let f ( t ) = ∑ i α i t i . {\textstyle f(t)=\sum _{i}\alpha _{i}t^{i}.} Then f ( A ) = ∑ α i ( S − 1 U S ) i = ∑ α i S − 1 U S S − 1 U S ⋯ S − 1 U S = ∑ α i S − 1 U i S = S − 1 ( ∑ α i U i ) S = S − 1 f ( U ) S . {\displaystyle f(A)=\textstyle \sum \alpha _{i}(S^{-1}US)^{i}=\textstyle \sum \alpha _{i}S^{-1}USS^{-1}US\cdots S^{-1}US=\textstyle \sum \alpha _{i}S^{-1}U^{i}S=S^{-1}(\textstyle \sum \alpha _{i}U^{i})S=S^{-1}f(U)S.} For an upper triangular matrix U {\displaystyle U} with diagonal λ 1 , … , λ n , {\displaystyle \lambda _{1},\dots ,\lambda _{n},}

2632-1410: Is a rotation matrix . Standard examples of such usage is the exponential map from the Lie algebra of a matrix Lie group into the group. It is given by a matrix exponential , exp : g → G ; t X ↦ e t X = ∑ n = 0 ∞ t n X n n ! = I + t X + t 2 X 2 2 + ⋯ , t ∈ R , X ∈ g . {\displaystyle \exp :{\mathfrak {g}}\rightarrow G;\qquad tX\mapsto e^{tX}=\sum _{n=0}^{\infty }{\frac {t^{n}X^{n}}{n!}}=I+tX+{\frac {t^{2}X^{2}}{2}}+\cdots ,t\in \mathbb {R} ,X\in {\mathfrak {g}}.} Such expressions have long been known for SU(2) , e i ( θ / 2 ) ( n ^ ⋅ σ ) = I 2 cos ⁡ θ 2 + i ( n ^ ⋅ σ ) sin ⁡ θ 2 , {\displaystyle e^{i(\theta /2)({\hat {\mathbf {n} }}\cdot \sigma )}=I_{2}\cos {\frac {\theta }{2}}+i({\hat {\mathbf {n} }}\cdot \sigma )\sin {\frac {\theta }{2}},} where

2726-937: Is a non-zero linear combination of the α 1 n 1 ⋯ α k n k {\displaystyle \alpha _{1}^{n_{1}}\cdots \alpha _{k}^{n_{k}}} we can compute the minimal polynomial of α {\displaystyle \alpha } by finding a matrix representing the Q {\displaystyle \mathbb {Q} } - linear transformation ⋅ α : Q [ α 1 , … , α k ] → Q [ α 1 , … , α k ] {\displaystyle \cdot \alpha :\mathbb {Q} [\alpha _{1},\ldots ,\alpha _{k}]\to \mathbb {Q} [\alpha _{1},\ldots ,\alpha _{k}]} If we call this transformation matrix A {\displaystyle A} , then we can find

2820-961: Is an eigenvalue of a square matrix A {\displaystyle A} with eigenvector v , {\displaystyle \mathbf {v} ,} then λ k {\displaystyle \lambda ^{k}} is an eigenvalue of A k {\displaystyle A^{k}} because A k v = A k − 1 A v = λ A k − 1 v = ⋯ = λ k v . {\displaystyle A^{k}{\textbf {v}}=A^{k-1}A{\textbf {v}}=\lambda A^{k-1}{\textbf {v}}=\dots =\lambda ^{k}{\textbf {v}}.} The multiplicities can be shown to agree as well, and this generalizes to any polynomial in place of x k {\displaystyle x^{k}} : Theorem  —  Let A {\displaystyle A} be

2914-488: Is apparent from the general formula for c n − k , expressed in terms of Bell polynomials, that the expressions − tr ⁡ ( A ) and 1 2 ( tr ⁡ ( A ) 2 − tr ⁡ ( A 2 ) ) {\displaystyle -\operatorname {tr} (A)\quad {\text{and}}\quad {\tfrac {1}{2}}(\operatorname {tr} (A)^{2}-\operatorname {tr} (A^{2}))} always give

Cayley–Hamilton theorem - Misplaced Pages Continue

3008-502: Is equal to A B {\displaystyle AB} bordered by n − m {\displaystyle n-m} rows and columns of zeros. The result follows from the case of square matrices, by comparing the characteristic polynomials of A ′ B ′ {\displaystyle A^{\prime }B^{\prime }} and A B . {\displaystyle AB.} If λ {\displaystyle \lambda }

3102-765: Is even. To compute the characteristic polynomial of the matrix A = ( 2 1 − 1 0 ) . {\displaystyle A={\begin{pmatrix}2&1\\-1&0\end{pmatrix}}.} the determinant of the following is computed: t I − A = ( t − 2 − 1 1 t − 0 ) {\displaystyle tI-A={\begin{pmatrix}t-2&-1\\1&t-0\end{pmatrix}}} and found to be ( t − 2 ) t − 1 ( − 1 ) = t 2 − 2 t + 1 , {\displaystyle (t-2)t-1(-1)=t^{2}-2t+1\,\!,}

3196-1291: Is led to an expression for the inverse of A as a trace identity, A − 1 = ( − 1 ) n − 1 det A ( A n − 1 + c n − 1 A n − 2 + ⋯ + c 1 I n ) , = 1 det A ∑ k = 0 n − 1 ( − 1 ) n + k − 1 A n − k − 1 k ! B k ( s 1 , − 1 ! s 2 , 2 ! s 3 , … , ( − 1 ) k − 1 ( k − 1 ) ! s k ) . {\displaystyle {\begin{aligned}A^{-1}&={\frac {(-1)^{n-1}}{\det A}}(A^{n-1}+c_{n-1}A^{n-2}+\cdots +c_{1}I_{n}),\\[5pt]&={\frac {1}{\det A}}\sum _{k=0}^{n-1}(-1)^{n+k-1}{\frac {A^{n-k-1}}{k!}}B_{k}(s_{1},-1!s_{2},2!s_{3},\ldots ,(-1)^{k-1}(k-1)!s_{k}).\end{aligned}}} Another method for obtaining these coefficients c k for

3290-973: Is similar to a triangular matrix if and only if its characteristic polynomial can be completely factored into linear factors over K {\displaystyle K} (the same is true with the minimal polynomial instead of the characteristic polynomial). In this case A {\displaystyle A} is similar to a matrix in Jordan normal form . If A {\displaystyle A} and B {\displaystyle B} are two square n × n {\displaystyle n\times n} matrices then characteristic polynomials of A B {\displaystyle AB} and B A {\displaystyle BA} coincide: p A B ( t ) = p B A ( t ) . {\displaystyle p_{AB}(t)=p_{BA}(t).\,} When A {\displaystyle A}

3384-428: Is some quotient polynomial and r ( x ) is a remainder polynomial such that 0 ≤ deg r ( x ) < n . By the Cayley–Hamilton theorem, replacing x by the matrix A gives p ( A ) = 0 , so one has f ( A ) = r ( A ) . {\displaystyle f(A)=r(A).} Thus, the analytic function of the matrix A can be expressed as a matrix polynomial of degree less than n . Let

3478-616: Is the identity matrix , and v ≠ 0 {\displaystyle \mathbf {v} \neq \mathbf {0} } (although the zero vector satisfies this equation for every λ , {\displaystyle \lambda ,} it is not considered an eigenvector). It follows that the matrix ( λ I − A ) {\displaystyle (\lambda I-A)} must be singular , and its determinant det ( λ I − A ) = 0 {\displaystyle \det(\lambda I-A)=0} must be zero. In other words,

3572-447: Is the trace of A . {\displaystyle A.} (The signs given here correspond to the formal definition given in the previous section; for the alternative definition these would instead be det ( A ) {\displaystyle \det(A)} and (−1) tr( A ) respectively. ) For a 2 × 2 {\displaystyle 2\times 2} matrix A , {\displaystyle A,}

3666-531: Is the trace of the k {\displaystyle k} th exterior power of A , {\displaystyle A,} which has dimension ( n k ) . {\textstyle {\binom {n}{k}}.} This trace may be computed as the sum of all principal minors of A {\displaystyle A} of size k . {\displaystyle k.} The recursive Faddeev–LeVerrier algorithm computes these coefficients more efficiently. When

3760-754: Is the trace of the matrix A . Thus, we can express c i in terms of the trace of powers of A . In general, the formula for the coefficients c i is given in terms of complete exponential Bell polynomials as c n − k = ( − 1 ) k k ! B k ( s 1 , − 1 ! s 2 , 2 ! s 3 , … , ( − 1 ) k − 1 ( k − 1 ) ! s k ) . {\displaystyle c_{n-k}={\frac {(-1)^{k}}{k!}}B_{k}(s_{1},-1!s_{2},2!s_{3},\ldots ,(-1)^{k-1}(k-1)!s_{k}).} In particular,

3854-468: Is the given matrix—not a variable, unlike λ {\displaystyle \lambda } —so p A ( A ) {\displaystyle p_{A}(A)} is a constant rather than a function.) The Cayley–Hamilton theorem states that this polynomial expression is equal to the zero matrix , which is to say that p A ( A ) = 0 ; {\displaystyle p_{A}(A)=\mathbf {0} ;} that is,

Cayley–Hamilton theorem - Misplaced Pages Continue

3948-521: Is the matrix with all entries equal to 0 K {\displaystyle 0_{K}\,} , where 0 K {\displaystyle 0_{K}} is the additive identity in K. The zero matrix is the additive identity in K m , n {\displaystyle K_{m,n}\,} . That is, for all A ∈ K m , n {\displaystyle A\in K_{m,n}\,} it satisfies

4042-513: Is the polynomial defined by p A ( t ) = det ( t I − A ) {\displaystyle p_{A}(t)=\det(tI-A)} where I {\displaystyle I} denotes the n × n {\displaystyle n\times n} identity matrix . Some authors define the characteristic polynomial to be det ( A − t I ) . {\displaystyle \det(A-tI).} That polynomial differs from

4136-676: Is upper triangular with diagonal f ( λ 1 ) , … , f ( λ n ) . {\displaystyle f\left(\lambda _{1}\right),\dots ,f\left(\lambda _{n}\right).} Therefore, the eigenvalues of f ( U ) {\displaystyle f(U)} are f ( λ 1 ) , … , f ( λ n ) . {\displaystyle f(\lambda _{1}),\dots ,f(\lambda _{n}).} Since f ( A ) = S − 1 f ( U ) S {\displaystyle f(A)=S^{-1}f(U)S}

4230-493: The Mercator series , p ( λ ) = λ n exp ⁡ ( − tr ⁡ ∑ m = 1 ∞ ( A λ ) m m ) , {\displaystyle p(\lambda )=\lambda ^{n}\exp \left(-\operatorname {tr} \sum _{m=1}^{\infty }{({A \over \lambda })^{m} \over m}\right),} where

4324-1662: The characteristic of the field of the coefficients is 0 , {\displaystyle 0,} each such trace may alternatively be computed as a single determinant, that of the k × k {\displaystyle k\times k} matrix, tr ⁡ ( ⋀ k A ) = 1 k ! | tr ⁡ A k − 1 0 ⋯ 0 tr ⁡ A 2 tr ⁡ A k − 2 ⋯ 0 ⋮ ⋮ ⋱ ⋮ tr ⁡ A k − 1 tr ⁡ A k − 2 ⋯ 1 tr ⁡ A k tr ⁡ A k − 1 ⋯ tr ⁡ A |   . {\displaystyle \operatorname {tr} \left(\textstyle \bigwedge ^{k}A\right)={\frac {1}{k!}}{\begin{vmatrix}\operatorname {tr} A&k-1&0&\cdots &0\\\operatorname {tr} A^{2}&\operatorname {tr} A&k-2&\cdots &0\\\vdots &\vdots &&\ddots &\vdots \\\operatorname {tr} A^{k-1}&\operatorname {tr} A^{k-2}&&\cdots &1\\\operatorname {tr} A^{k}&\operatorname {tr} A^{k-1}&&\cdots &\operatorname {tr} A\end{vmatrix}}~.} The Cayley–Hamilton theorem states that replacing t {\displaystyle t} by A {\displaystyle A} in

4418-417: The characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots . It has the determinant and the trace of the matrix among its coefficients. The characteristic polynomial of an endomorphism of a finite-dimensional vector space is the characteristic polynomial of the matrix of that endomorphism over any basis (that is,

4512-503: The elementary symmetric polynomials of the eigenvalues of A . Using Newton identities , the elementary symmetric polynomials can in turn be expressed in terms of power sum symmetric polynomials of the eigenvalues: s k = ∑ i = 1 n λ i k = tr ⁡ ( A k ) , {\displaystyle s_{k}=\sum _{i=1}^{n}\lambda _{i}^{k}=\operatorname {tr} (A^{k}),} where tr( A )

4606-483: The roots of p A ( t ) {\displaystyle p_{A}(t)} (this also holds for the minimal polynomial of A , {\displaystyle A,} but its degree may be less than n {\displaystyle n} ). All coefficients of the characteristic polynomial are polynomial expressions in the entries of the matrix. In particular its constant coefficient of t 0 {\displaystyle t^{0}}

4700-701: The σ are the Pauli matrices and for SO(3) , e i θ ( n ^ ⋅ J ) = I 3 + i ( n ^ ⋅ J ) sin ⁡ θ + ( n ^ ⋅ J ) 2 ( cos ⁡ θ − 1 ) , {\displaystyle e^{i\theta ({\hat {\mathbf {n} }}\cdot \mathbf {J} )}=I_{3}+i({\hat {\mathbf {n} }}\cdot \mathbf {J} )\sin \theta +({\hat {\mathbf {n} }}\cdot \mathbf {J} )^{2}(\cos \theta -1),} which

4794-1089: The algebraic multiplicity of λ {\displaystyle \lambda } in f ( A ) {\displaystyle f(A)} equals the sum of algebraic multiplicities of λ ′ {\displaystyle \lambda '} in A {\displaystyle A} over λ ′ {\displaystyle \lambda '} such that f ( λ ′ ) = λ . {\displaystyle f(\lambda ')=\lambda .} In particular, tr ⁡ ( f ( A ) ) = ∑ i = 1 n f ( λ i ) {\displaystyle \operatorname {tr} (f(A))=\textstyle \sum _{i=1}^{n}f(\lambda _{i})} and det ⁡ ( f ( A ) ) = ∏ i = 1 n f ( λ i ) . {\displaystyle \operatorname {det} (f(A))=\textstyle \prod _{i=1}^{n}f(\lambda _{i}).} Here

SECTION 50

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4888-1028: The assumption that p A ( t ) {\displaystyle p_{A}(t)} has a factorization into linear factors is not always true, unless the matrix is over an algebraically closed field such as the complex numbers. This proof only applies to matrices and polynomials over complex numbers (or any algebraically closed field). In that case, the characteristic polynomial of any square matrix can be always factorized as p A ( t ) = ( t − λ 1 ) ( t − λ 2 ) ⋯ ( t − λ n ) {\displaystyle p_{A}(t)=\left(t-\lambda _{1}\right)\left(t-\lambda _{2}\right)\cdots \left(t-\lambda _{n}\right)} where λ 1 , λ 2 , … , λ n {\displaystyle \lambda _{1},\lambda _{2},\ldots ,\lambda _{n}} are

4982-750: The bottom by n − m {\displaystyle n-m} rows of zeros, and B {\displaystyle B} on the right, by, n − m {\displaystyle n-m} columns of zeros, one gets two n × n {\displaystyle n\times n} matrices A ′ {\displaystyle A^{\prime }} and B ′ {\displaystyle B^{\prime }} such that B ′ A ′ = B A {\displaystyle B^{\prime }A^{\prime }=BA} and A ′ B ′ {\displaystyle A^{\prime }B^{\prime }}

5076-470: The case when F {\displaystyle F} is just a commutative ring . Garibaldi (2004) defines the characteristic polynomial for elements of an arbitrary finite-dimensional ( associative , but not necessarily commutative) algebra over a field F {\displaystyle F} and proves the standard properties of the characteristic polynomial in this generality. Zero matrix In mathematics , particularly linear algebra ,

5170-643: The characteristic polynomial p A {\displaystyle p_{A}} is an annihilating polynomial for A . {\displaystyle A.} One use for the Cayley–Hamilton theorem is that it allows A to be expressed as a linear combination of the lower matrix powers of A : A n = − c n − 1 A n − 1 − ⋯ − c 1 A − c 0 I n . {\displaystyle A^{n}=-c_{n-1}A^{n-1}-\cdots -c_{1}A-c_{0}I_{n}.} When

5264-453: The characteristic polynomial (interpreting the resulting powers as matrix powers, and the constant term c {\displaystyle c} as c {\displaystyle c} times the identity matrix) yields the zero matrix. Informally speaking, every matrix satisfies its own characteristic equation. This statement is equivalent to saying that the minimal polynomial of A {\displaystyle A} divides

5358-511: The characteristic polynomial can be written as − ( − 1 ) n det ( A ) I n = A ( A n − 1 + c n − 1 A n − 2 + ⋯ + c 1 I n ) , {\displaystyle -(-1)^{n}\det(A)I_{n}=A(A^{n-1}+c_{n-1}A^{n-2}+\cdots +c_{1}I_{n}),} and, by multiplying both sides by A (note −(−1) = (−1) ), one

5452-423: The characteristic polynomial does not depend on the choice of a basis ). The characteristic equation , also known as the determinantal equation , is the equation obtained by equating the characteristic polynomial to zero. In spectral graph theory , the characteristic polynomial of a graph is the characteristic polynomial of its adjacency matrix . In linear algebra , eigenvalues and eigenvectors play

5546-965: The characteristic polynomial is thus given by t 2 − tr ⁡ ( A ) t + det ( A ) . {\displaystyle t^{2}-\operatorname {tr} (A)t+\det(A).} Using the language of exterior algebra , the characteristic polynomial of an n × n {\displaystyle n\times n} matrix A {\displaystyle A} may be expressed as p A ( t ) = ∑ k = 0 n t n − k ( − 1 ) k tr ⁡ ( ⋀ k A ) {\displaystyle p_{A}(t)=\sum _{k=0}^{n}t^{n-k}(-1)^{k}\operatorname {tr} \left(\textstyle \bigwedge ^{k}A\right)} where tr ⁡ ( ⋀ k A ) {\textstyle \operatorname {tr} \left(\bigwedge ^{k}A\right)}

5640-593: The characteristic polynomial of A . {\displaystyle A.} Another example uses hyperbolic functions of a hyperbolic angle φ. For the matrix take A = ( cosh ⁡ ( φ ) sinh ⁡ ( φ ) sinh ⁡ ( φ ) cosh ⁡ ( φ ) ) . {\displaystyle A={\begin{pmatrix}\cosh(\varphi )&\sinh(\varphi )\\\sinh(\varphi )&\cosh(\varphi )\end{pmatrix}}.} Its characteristic polynomial

5734-428: The characteristic polynomial of A . {\displaystyle A.} Two similar matrices have the same characteristic polynomial. The converse however is not true in general: two matrices with the same characteristic polynomial need not be similar. The matrix A {\displaystyle A} and its transpose have the same characteristic polynomial. A {\displaystyle A}

SECTION 60

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5828-524: The characteristic polynomial of the matrix f ( A ) {\displaystyle f(A)} is given by p f ( A ) ( t ) = ( t − f ( λ 1 ) ) ( t − f ( λ 2 ) ) ⋯ ( t − f ( λ n ) ) . {\displaystyle p_{f(A)}(t)=(t-f(\lambda _{1}))(t-f(\lambda _{2}))\cdots (t-f(\lambda _{n})).} That is,

5922-619: The coefficients c k is deducible from Newton's identities or the Faddeev–LeVerrier algorithm . The Cayley–Hamilton theorem always provides a relationship between the powers of A (though not always the simplest one), which allows one to simplify expressions involving such powers, and evaluate them without having to compute the power A or any higher powers of A . As an example, for A = ( 1 2 3 4 ) {\displaystyle A={\begin{pmatrix}1&2\\3&4\end{pmatrix}}}

6016-744: The coefficients c n −1 of λ and c n −2 of λ in the characteristic polynomial of any n  ×  n matrix, respectively. So, for a 3 × 3 matrix A , the statement of the Cayley–Hamilton theorem can also be written as A 3 − ( tr ⁡ A ) A 2 + 1 2 ( ( tr ⁡ A ) 2 − tr ⁡ ( A 2 ) ) A − det ( A ) I 3 = O , {\displaystyle A^{3}-(\operatorname {tr} A)A^{2}+{\frac {1}{2}}\left((\operatorname {tr} A)^{2}-\operatorname {tr} (A^{2})\right)A-\det(A)I_{3}=O,} where

6110-588: The corresponding eigenvalue λ {\displaystyle \lambda } must satisfy the equation A v = λ v , {\displaystyle A\mathbf {v} =\lambda \mathbf {v} ,} or, equivalently (since λ v = λ I v {\displaystyle \lambda \mathbf {v} =\lambda I\mathbf {v} } ), ( λ I − A ) v = 0 {\displaystyle (\lambda I-A)\mathbf {v} =\mathbf {0} } where I {\displaystyle I}

6204-466: The desired identity is an equality between polynomials in t {\displaystyle t} and the coefficients of the matrices. Thus, to prove this equality, it suffices to prove that it is verified on a non-empty open subset (for the usual topology , or, more generally, for the Zariski topology ) of the space of all the coefficients. As the non-singular matrices form such an open subset of

6298-836: The determinant is c n −4 , 1 24 [ ( tr ⁡ A ) 4 − 6 tr ⁡ ( A 2 ) ( tr ⁡ A ) 2 + 3 ( tr ⁡ ( A 2 ) ) 2 + 8 tr ⁡ ( A 3 ) tr ⁡ ( A ) − 6 tr ⁡ ( A 4 ) ] , {\displaystyle {\tfrac {1}{24}}\!\left[(\operatorname {tr} A)^{4}-6\operatorname {tr} (A^{2})(\operatorname {tr} A)^{2}+3\left(\operatorname {tr} (A^{2})\right)^{2}+8\operatorname {tr} (A^{3})\operatorname {tr} (A)-6\operatorname {tr} (A^{4})\right],} and so on for larger matrices. The increasingly complex expressions for

6392-565: The determinant of A equals (−1) c 0 . Thus, the determinant can be written as the trace identity : det ( A ) = 1 n ! B n ( s 1 , − 1 ! s 2 , 2 ! s 3 , … , ( − 1 ) n − 1 ( n − 1 ) ! s n ) . {\displaystyle \det(A)={\frac {1}{n!}}B_{n}(s_{1},-1!s_{2},2!s_{3},\ldots ,(-1)^{n-1}(n-1)!s_{n}).} Likewise,

6486-674: The determinant of the 2 × 2 matrix, c 1 minus its trace, while its inverse is given by A − 1 = − 1 det A ( A + c 1 I 2 ) = − 2 ( A − tr ⁡ ( A ) I 2 ) ( tr ⁡ ( A ) ) 2 − tr ⁡ ( A 2 ) . {\displaystyle A^{-1}={\frac {-1}{\det A}}(A+c_{1}I_{2})={\frac {-2(A-\operatorname {tr} (A)I_{2})}{(\operatorname {tr} (A))^{2}-\operatorname {tr} (A^{2})}}.} It

6580-762: The eigenvalues are λ = ± i . As before, evaluating the function at the eigenvalues gives us the linear equations e = c 0 + i c 1 and e = c 0 − ic 1 ; the solution of which gives, c 0 = ( e + e )/2 = cos  t and c 1 = ( e − e )/2 i = sin  t . Thus, for this case, e A t = ( cos ⁡ t ) I 2 + ( sin ⁡ t ) A = ( cos ⁡ t sin ⁡ t − sin ⁡ t cos ⁡ t ) , {\displaystyle e^{At}=(\cos t)I_{2}+(\sin t)A={\begin{pmatrix}\cos t&\sin t\\-\sin t&\cos t\end{pmatrix}},} which

6674-463: The eigenvalues of A , {\displaystyle A,} possibly repeated. Moreover, the Jordan decomposition theorem guarantees that any square matrix A {\displaystyle A} can be decomposed as A = S − 1 U S , {\displaystyle A=S^{-1}US,} where S {\displaystyle S} is an invertible matrix and U {\displaystyle U}

6768-602: The eigenvalues of A are the roots of det ( x I − A ) , {\displaystyle \det(xI-A),} which is a monic polynomial in x of degree n if A is a n × n matrix. This polynomial is the characteristic polynomial of A . Consider an n × n {\displaystyle n\times n} matrix A . {\displaystyle A.} The characteristic polynomial of A , {\displaystyle A,} denoted by p A ( t ) , {\displaystyle p_{A}(t),}

6862-420: The equation There is exactly one zero matrix of any given dimension m × n (with entries from a given ring), so when the context is clear, one often refers to the zero matrix. In general, the zero element of a ring is unique, and is typically denoted by 0 without any subscript indicating the parent ring. Hence the examples above represent zero matrices over any ring. The zero matrix also represents

6956-1687: The exponential only needs be expanded to order λ , since p ( λ ) is of order n , the net negative powers of λ automatically vanishing by the C–H theorem. (Again, this requires a ring containing the rational numbers .) Differentiation of this expression with respect to λ allows one to express the coefficients of the characteristic polynomial for general n as determinants of m  ×  m matrices, c n − m = ( − 1 ) m m ! | tr ⁡ A m − 1 0 ⋯ tr ⁡ A 2 tr ⁡ A m − 2 ⋯ ⋮ ⋮ ⋮ tr ⁡ A m − 1 tr ⁡ A m − 2 ⋯ ⋯ 1 tr ⁡ A m tr ⁡ A m − 1 ⋯ ⋯ tr ⁡ A |   . {\displaystyle c_{n-m}={\frac {(-1)^{m}}{m!}}{\begin{vmatrix}\operatorname {tr} A&m-1&0&\cdots \\\operatorname {tr} A^{2}&\operatorname {tr} A&m-2&\cdots \\\vdots &\vdots &&&\vdots \\\operatorname {tr} A^{m-1}&\operatorname {tr} A^{m-2}&\cdots &\cdots &1\\\operatorname {tr} A^{m}&\operatorname {tr} A^{m-1}&\cdots &\cdots &\operatorname {tr} A\end{vmatrix}}~.} For instance,

7050-1389: The first few Bell polynomials are B 0 = 1, B 1 ( x 1 ) = x 1 , B 2 ( x 1 , x 2 ) = x 1 + x 2 , and B 3 ( x 1 , x 2 , x 3 ) = x 1 + 3 x 1 x 2 + x 3 . Using these to specify the coefficients c i of the characteristic polynomial of a 2 × 2 matrix yields c 2 = B 0 = 1 , c 1 = − 1 1 ! B 1 ( s 1 ) = − s 1 = − tr ⁡ ( A ) , c 0 = 1 2 ! B 2 ( s 1 , − 1 ! s 2 ) = 1 2 ( s 1 2 − s 2 ) = 1 2 ( ( tr ⁡ ( A ) ) 2 − tr ⁡ ( A 2 ) ) . {\displaystyle {\begin{aligned}c_{2}=B_{0}=1,\\[4pt]c_{1}={\frac {-1}{1!}}B_{1}(s_{1})=-s_{1}=-\operatorname {tr} (A),\\[4pt]c_{0}={\frac {1}{2!}}B_{2}(s_{1},-1!s_{2})={\frac {1}{2}}(s_{1}^{2}-s_{2})={\frac {1}{2}}((\operatorname {tr} (A))^{2}-\operatorname {tr} (A^{2})).\end{aligned}}} The coefficient c 0 gives

7144-645: The function were f ( A ) = sin At , then the coefficients would have been c 0 = (3 sin t − sin 3 t )/2 and c 1 = (sin 3 t − sin t )/2 ; hence sin ⁡ ( A t ) = c 0 I 2 + c 1 A = ( sin ⁡ t sin ⁡ 3 t − sin ⁡ t 0 sin ⁡ 3 t ) . {\displaystyle \sin(At)=c_{0}I_{2}+c_{1}A={\begin{pmatrix}\sin t&\sin 3t-\sin t\\0&\sin 3t\end{pmatrix}}.} As

7238-871: The linear equations cannot be solved uniquely. For such cases, for an eigenvalue λ with multiplicity m , the first m – 1 derivatives of p ( x ) vanish at the eigenvalue. This leads to the extra m – 1 linearly independent solutions d k f ( x ) d x k | x = λ = d k r ( x ) d x k | x = λ for  k = 1 , 2 , … , m − 1 , {\displaystyle \left.{\frac {\mathrm {d} ^{k}f(x)}{\mathrm {d} x^{k}}}\right|_{x=\lambda }=\left.{\frac {\mathrm {d} ^{k}r(x)}{\mathrm {d} x^{k}}}\right|_{x=\lambda }\qquad {\text{for }}k=1,2,\ldots ,m-1,} which, combined with others, yield

7332-419: The matrix U i {\displaystyle U^{i}} is upper triangular with diagonal λ 1 i , … , λ n i {\displaystyle \lambda _{1}^{i},\dots ,\lambda _{n}^{i}} in U i , {\displaystyle U^{i},} and hence f ( U ) {\displaystyle f(U)}

7426-740: The matrix ( λ I n − A ) {\displaystyle (\lambda I_{n}-A)} is either constant or linear in λ , the determinant of ( λ I n − A ) {\displaystyle (\lambda I_{n}-A)} is a degree - n monic polynomial in λ , so it can be written as p A ( λ ) = λ n + c n − 1 λ n − 1 + ⋯ + c 1 λ + c 0 . {\displaystyle p_{A}(\lambda )=\lambda ^{n}+c_{n-1}\lambda ^{n-1}+\cdots +c_{1}\lambda +c_{0}.} By replacing

7520-470: The matrix as the context sees fit. Some examples of zero matrices are The set of m × n {\displaystyle m\times n} matrices with entries in a ring K forms a ring K m , n {\displaystyle K_{m,n}} . The zero matrix 0 K m , n {\displaystyle 0_{K_{m,n}}\,} in K m , n {\displaystyle K_{m,n}\,}

7614-440: The matrix power as the sum of two terms. In fact, matrix power of any order k can be written as a matrix polynomial of degree at most n − 1 , where n is the size of a square matrix. This is an instance where Cayley–Hamilton theorem can be used to express a matrix function, which we will discuss below systematically. Given an analytic function f ( x ) = ∑ k = 0 ∞

7708-407: The minimal polynomial by applying the Cayley–Hamilton theorem to A {\displaystyle A} . The Cayley–Hamilton theorem is an immediate consequence of the existence of the Jordan normal form for matrices over algebraically closed fields , see Jordan normal form § Cayley–Hamilton theorem . In this section, direct proofs are presented. As the examples above show, obtaining

7802-1049: The negative of coefficient c n −3 of λ in the general case, as seen below. Similarly, one can write for a 4 × 4 matrix A , A 4 − ( tr ⁡ A ) A 3 + 1 2 [ ( tr ⁡ A ) 2 − tr ⁡ ( A 2 ) ] A 2 − 1 6 [ ( tr ⁡ A ) 3 − 3 tr ⁡ ( A 2 ) ( tr ⁡ A ) + 2 tr ⁡ ( A 3 ) ] A + det ( A ) I 4 = O , {\displaystyle A^{4}-(\operatorname {tr} A)A^{3}+{\tfrac {1}{2}}\left[(\operatorname {tr} A)^{2}-\operatorname {tr} (A^{2})\right]A^{2}-{\tfrac {1}{6}}\left[(\operatorname {tr} A)^{3}-3\operatorname {tr} (A^{2})(\operatorname {tr} A)+2\operatorname {tr} (A^{3})\right]A+\det(A)I_{4}=O,} where, now,

7896-409: The one defined here by a sign ( − 1 ) n , {\displaystyle (-1)^{n},} so it makes no difference for properties like having as roots the eigenvalues of A {\displaystyle A} ; however the definition above always gives a monic polynomial , whereas the alternative definition is monic only when n {\displaystyle n}

7990-908: The remainder polynomial be r ( x ) = c 0 + c 1 x + ⋯ + c n − 1 x n − 1 . {\displaystyle r(x)=c_{0}+c_{1}x+\cdots +c_{n-1}x^{n-1}.} Since p ( λ ) = 0 , evaluating the function f ( x ) at the n eigenvalues of A yields f ( λ i ) = r ( λ i ) = c 0 + c 1 λ i + ⋯ + c n − 1 λ i n − 1 , for  i = 1 , 2 , . . . , n . {\displaystyle f(\lambda _{i})=r(\lambda _{i})=c_{0}+c_{1}\lambda _{i}+\cdots +c_{n-1}\lambda _{i}^{n-1},\qquad {\text{for }}i=1,2,...,n.} This amounts to

8084-699: The required n equations to solve for c i . Finding a polynomial that passes through the points ( λ i ,   f  ( λ i )) is essentially an interpolation problem , and can be solved using Lagrange or Newton interpolation techniques, leading to Sylvester's formula . For example, suppose the task is to find the polynomial representation of f ( A ) = e A t w h e r e A = ( 1 2 0 3 ) . {\displaystyle f(A)=e^{At}\qquad \mathrm {where} \qquad A={\begin{pmatrix}1&2\\0&3\end{pmatrix}}.} The characteristic polynomial

8178-426: The result for 3 × 3 and smaller matrices, but only published a proof for the 2 × 2 case. As for n  ×  n matrices, Cayley stated “..., I have not thought it necessary to undertake the labor of a formal proof of the theorem in the general case of a matrix of any degree”. The general case was first proved by Ferdinand Frobenius in 1878. For a 1 × 1 matrix A = (

8272-1114: The right-hand side designates a 3 × 3 matrix with all entries reduced to zero. Likewise, this determinant in the n = 3 case, is now det ( A ) = 1 3 ! B 3 ( s 1 , − 1 ! s 2 , 2 ! s 3 ) = 1 6 ( s 1 3 + 3 s 1 ( − s 2 ) + 2 s 3 ) = 1 6 [ ( tr ⁡ A ) 3 − 3 tr ⁡ ( A 2 ) ( tr ⁡ A ) + 2 tr ⁡ ( A 3 ) ] . {\displaystyle {\begin{aligned}\det(A)&={\frac {1}{3!}}B_{3}(s_{1},-1!s_{2},2!s_{3})={\frac {1}{6}}(s_{1}^{3}+3s_{1}(-s_{2})+2s_{3})\\[5pt]&={\frac {1}{6}}\left[(\operatorname {tr} A)^{3}-3\operatorname {tr} (A^{2})(\operatorname {tr} A)+2\operatorname {tr} (A^{3})\right].\end{aligned}}} This expression gives

8366-435: The ring is a field , the Cayley–Hamilton theorem is equivalent to the statement that the minimal polynomial of a square matrix divides its characteristic polynomial. A special case of the theorem was first proved by Hamilton in 1853 in terms of inverses of linear functions of quaternions . This corresponds to the special case of certain 4 × 4 real or 2 × 2 complex matrices. Cayley in 1858 stated

8460-472: The scalar variable λ with the matrix A , one can define an analogous matrix polynomial expression, p A ( A ) = A n + c n − 1 A n − 1 + ⋯ + c 1 A + c 0 I n . {\displaystyle p_{A}(A)=A^{n}+c_{n-1}A^{n-1}+\cdots +c_{1}A+c_{0}I_{n}.} (Here, A {\displaystyle A}

8554-408: The space of all matrices, this proves the result. More generally, if A {\displaystyle A} is a matrix of order m × n {\displaystyle m\times n} and B {\displaystyle B} is a matrix of order n × m , {\displaystyle n\times m,} then A B {\displaystyle AB}

8648-902: The standard representation of these groups. They require knowledge of (some of) the eigenvalues of the matrix to exponentiate. For SU(2) (and hence for SO(3) ), closed expressions have been obtained for all irreducible representations, i.e. of any spin. The Cayley–Hamilton theorem is an effective tool for computing the minimal polynomial of algebraic integers . For example, given a finite extension Q [ α 1 , … , α k ] {\displaystyle \mathbb {Q} [\alpha _{1},\ldots ,\alpha _{k}]} of Q {\displaystyle \mathbb {Q} } and an algebraic integer α ∈ Q [ α 1 , … , α k ] {\displaystyle \alpha \in \mathbb {Q} [\alpha _{1},\ldots ,\alpha _{k}]} which

8742-555: The statement of the Cayley–Hamilton theorem for an n  ×  n matrix A = ( a i j ) i , j = 1 n {\displaystyle A=\left(a_{ij}\right)_{i,j=1}^{n}} requires two steps: first the coefficients c i of the characteristic polynomial are determined by development as a polynomial in t of the determinant p ( t ) = det ( t I n − A ) = | t −

8836-1921: The theorem gives A 2 = 5 A + 2 I 2 . {\displaystyle A^{2}=5A+2I_{2}\,.} Then, to calculate A , observe A 3 = ( 5 A + 2 I 2 ) A = 5 A 2 + 2 A = 5 ( 5 A + 2 I 2 ) + 2 A = 27 A + 10 I 2 , A 4 = A 3 A = ( 27 A + 10 I 2 ) A = 27 A 2 + 10 A = 27 ( 5 A + 2 I 2 ) + 10 A = 145 A + 54 I 2 . {\displaystyle {\begin{aligned}A^{3}&=(5A+2I_{2})A=5A^{2}+2A=5(5A+2I_{2})+2A=27A+10I_{2},\\[1ex]A^{4}&=A^{3}A=(27A+10I_{2})A=27A^{2}+10A=27(5A+2I_{2})+10A=145A+54I_{2}\,.\end{aligned}}} Likewise, A − 1 = 1 2 ( A − 5 I 2 )   . A − 2 = A − 1 A − 1 = 1 4 ( A 2 − 10 A + 25 I 2 ) = 1 4 ( ( 5 A + 2 I 2 ) − 10 A + 25 I 2 ) = 1 4 ( − 5 A + 27 I 2 )   . {\displaystyle {\begin{aligned}A^{-1}&={\frac {1}{2}}\left(A-5I_{2}\right)~.\\[1ex]A^{-2}&=A^{-1}A^{-1}={\frac {1}{4}}\left(A^{2}-10A+25I_{2}\right)={\frac {1}{4}}\left((5A+2I_{2})-10A+25I_{2}\right)={\frac {1}{4}}\left(-5A+27I_{2}\right)~.\end{aligned}}} Notice that we have been able to write

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