Misplaced Pages

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50-469: Cracovians introduced the idea of using the transpose of A , A , and multiplying the columns of A by the column x . This amounts to the definition of a new type of matrix multiplication denoted here by '∧'. Thus x ∧ A = b = A x . The Cracovian product of two matrices, say A and B , is defined by A ∧ B = B A , where B and A are assumed compatible for the common ( Cayley ) type of matrix multiplication. Since ( AB ) = B A ,

100-405: A skew-symmetric matrix ; that is, A is skew-symmetric if A square complex matrix whose transpose is equal to the matrix with every entry replaced by its complex conjugate (denoted here with an overline) is called a Hermitian matrix (equivalent to the matrix being equal to its conjugate transpose ); that is, A is Hermitian if A square complex matrix whose transpose is equal to

150-492: A complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign . That is, if a {\displaystyle a} and b {\displaystyle b} are real numbers then the complex conjugate of a + b i {\displaystyle a+bi} is a − b i . {\displaystyle a-bi.} The complex conjugate of z {\displaystyle z}

200-496: A real structure . As the involution φ {\displaystyle \varphi } is antilinear , it cannot be the identity map on V . {\displaystyle V.} Of course, φ {\textstyle \varphi } is a R {\textstyle \mathbb {R} } -linear transformation of V , {\textstyle V,} if one notes that every complex space V {\displaystyle V} has

250-407: A scalar . If A is an m × n matrix and A is its transpose, then the result of matrix multiplication with these two matrices gives two square matrices: A A is m × m and A A is n × n . Furthermore, these products are symmetric matrices . Indeed, the matrix product A A has entries that are the inner product of a row of A with a column of A . But

300-412: A topological vector space (TVS) X is denoted by X ' . If X and Y are TVSs then a linear map u  : X → Y is weakly continuous if and only if u ( Y ' ) ⊆ X ' , in which case we let u  : Y ' → X ' denote the restriction of u to Y ' . The map u is called the transpose of u . If the matrix A describes

350-421: A vinculum , avoids confusion with the notation for the conjugate transpose of a matrix , which can be thought of as a generalization of the complex conjugate. The second is preferred in physics , where dagger (†) is used for the conjugate transpose, as well as electrical engineering and computer engineering , where bar notation can be confused for the logical negation ("NOT") Boolean algebra symbol, while

400-576: A conjugation for quaternions and split-quaternions : the conjugate of a + b i + c j + d k {\textstyle a+bi+cj+dk} is a − b i − c j − d k . {\textstyle a-bi-cj-dk.} All these generalizations are multiplicative only if the factors are reversed: ( z w ) ∗ = w ∗ z ∗ . {\displaystyle {\left(zw\right)}^{*}=w^{*}z^{*}.} Since

450-475: A fixed complex unit u = e i b , {\displaystyle u=e^{ib},} the equation z − z 0 z ¯ − z 0 ¯ = u 2 {\displaystyle {\frac {z-z_{0}}{{\overline {z}}-{\overline {z_{0}}}}}=u^{2}} determines the line through z 0 {\displaystyle z_{0}} parallel to

500-405: A linear map with respect to bases of V and W , then the matrix A describes the transpose of that linear map with respect to the dual bases . Every linear map to the dual space u  : X → X defines a bilinear form B  : X × X → F , with the relation B ( x , y ) = u ( x )( y ) . By defining the transpose of this bilinear form as the bilinear form B defined by

550-683: A negative real number  {\displaystyle \ln \left({\overline {z}}\right)={\overline {\ln(z)}}{\text{ if }}z{\text{ is not zero or a negative real number }}} If p {\displaystyle p} is a polynomial with real coefficients and p ( z ) = 0 , {\displaystyle p(z)=0,} then p ( z ¯ ) = 0 {\displaystyle p\left({\overline {z}}\right)=0} as well. Thus, non-real roots of real polynomials occur in complex conjugate pairs ( see Complex conjugate root theorem ). In general, if φ {\displaystyle \varphi }

SECTION 10

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600-426: A number of circumstances in which it is necessary or desirable to physically reorder a matrix in memory to its transposed ordering. For example, with a matrix stored in row-major order , the rows of the matrix are contiguous in memory and the columns are discontiguous. If repeated operations need to be performed on the columns, for example in a fast Fourier transform algorithm, transposing the matrix in memory (to make

650-400: A real form obtained by taking the same vectors as in the original space and restricting the scalars to be real. The above properties actually define a real structure on the complex vector space V . {\displaystyle V.} One example of this notion is the conjugate transpose operation of complex matrices defined above. However, on generic complex vector spaces, there

700-400: A root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root . The complex conjugate of a complex number z {\displaystyle z} is written as z ¯ {\displaystyle {\overline {z}}} or z ∗ . {\displaystyle z^{*}.} The first notation,

750-441: A vertical column and a horizontal row in the matrix notation.) It also sped up computer calculations, because both factors' elements were used in a similar order, which was more compatible with the sequential access memory in computers of those times — mostly magnetic tape memory and drum memory . Use of Cracovians in astronomy faded as computers with bigger random access memory came into general use. Any modern reference to them

800-424: Is r e − i φ . {\displaystyle re^{-i\varphi }.} This can be shown using Euler's formula . The product of a complex number and its conjugate is a real number: a 2 + b 2 {\displaystyle a^{2}+b^{2}}  (or  r 2 {\displaystyle r^{2}} in polar coordinates ). If

850-457: Is bijective and compatible with the arithmetical operations, and hence is a field automorphism . As it keeps the real numbers fixed, it is an element of the Galois group of the field extension C / R . {\displaystyle \mathbb {C} /\mathbb {R} .} This Galois group has only two elements: σ {\displaystyle \sigma } and

900-783: Is a holomorphic function whose restriction to the real numbers is real-valued, and φ ( z ) {\displaystyle \varphi (z)} and φ ( z ¯ ) {\displaystyle \varphi ({\overline {z}})} are defined, then φ ( z ¯ ) = φ ( z ) ¯ . {\displaystyle \varphi \left({\overline {z}}\right)={\overline {\varphi (z)}}.\,\!} The map σ ( z ) = z ¯ {\displaystyle \sigma (z)={\overline {z}}} from C {\displaystyle \mathbb {C} } to C {\displaystyle \mathbb {C} }

950-447: Is a homeomorphism (where the topology on C {\displaystyle \mathbb {C} } is taken to be the standard topology) and antilinear , if one considers C {\displaystyle \mathbb {C} } as a complex vector space over itself. Even though it appears to be a well-behaved function, it is not holomorphic ; it reverses orientation whereas holomorphic functions locally preserve orientation. It

1000-481: Is a linear map between vector spaces X and Y , we define g as the adjoint of u if g  : Y → X satisfies These bilinear forms define an isomorphism between X and X , and between Y and Y , resulting in an isomorphism between the transpose and adjoint of u . The matrix of the adjoint of a map is the transposed matrix only if the bases are orthonormal with respect to their bilinear forms. In this context, many authors however, use

1050-412: Is a line through the origin and perpendicular to r , {\displaystyle {r},} since the real part of z ⋅ r ¯ {\displaystyle z\cdot {\overline {r}}} is zero only when the cosine of the angle between z {\displaystyle z} and r {\displaystyle {r}} is zero. Similarly, for

SECTION 20

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1100-423: Is equal to its complex conjugate if its imaginary part is zero, that is, if the number is real. In other words, real numbers are the only fixed points of conjugation. Conjugation does not change the modulus of a complex number: | z ¯ | = | z | . {\displaystyle \left|{\overline {z}}\right|=|z|.} Conjugation is an involution , that is,

1150-458: Is given, its conjugate is sufficient to reproduce the parts of the z {\displaystyle z} -variable: Furthermore, z ¯ {\displaystyle {\overline {z}}} can be used to specify lines in the plane: the set { z : z r ¯ + z ¯ r = 0 } {\displaystyle \left\{z:z{\overline {r}}+{\overline {z}}r=0\right\}}

1200-552: Is in connection with their non-associative multiplication. Named for recognition of the City of Cracow . In R the desired effect can be achieved via the crossprod() function. Specifically, the Cracovian product of matrices A and B can be obtained as crossprod(B, A) . Matrix transpose In linear algebra , the transpose of a matrix is an operator which flips a matrix over its diagonal; that is, it switches

1250-404: Is non-trivial to implement in-place. Therefore, efficient in-place matrix transposition has been the subject of numerous research publications in computer science , starting in the late 1950s, and several algorithms have been developed. As the main use of matrices is to represent linear maps between finite-dimensional vector spaces , the transpose is an operation on matrices that may be seen as

1300-429: Is often denoted as z ¯ {\displaystyle {\overline {z}}} or z ∗ {\displaystyle z^{*}} . In polar form , if r {\displaystyle r} and φ {\displaystyle \varphi } are real numbers then the conjugate of r e i φ {\displaystyle re^{i\varphi }}

1350-404: Is that no parentheses are needed when exponents are involved: as ( A ) = ( A ) , notation A is not ambiguous. In this article this confusion is avoided by never using the symbol T as a variable name. A square matrix whose transpose is equal to itself is called a symmetric matrix ; that is, A is symmetric if A square matrix whose transpose is equal to its negative is called

1400-403: The basis choice. Let X denote the algebraic dual space of an R - module X . Let X and Y be R -modules. If u  : X → Y is a linear map , then its algebraic adjoint or dual , is the map u  : Y → X defined by f ↦ f ∘ u . The resulting functional u ( f ) is called the pullback of f by u . The following relation characterizes

1450-406: The conjugate transpose of A . {\textstyle \mathbf {A} .} Taking the conjugate transpose (or adjoint) of complex matrices generalizes complex conjugation. Even more general is the concept of adjoint operator for operators on (possibly infinite-dimensional) complex Hilbert spaces . All this is subsumed by the *-operations of C*-algebras . One may also define

1500-460: The multiplicative inverse of a complex number given in rectangular coordinates: z − 1 = z ¯ | z | 2 ,  for all  z ≠ 0. {\displaystyle z^{-1}={\frac {\overline {z}}{{\left|z\right|}^{2}}},\quad {\text{ for all }}z\neq 0.} Conjugation is commutative under composition with exponentiation to integer powers, with

1550-406: The algebraic adjoint of u where ⟨•, •⟩ is the natural pairing (i.e. defined by ⟨ h , z ⟩  := h ( z ) ). This definition also applies unchanged to left modules and to vector spaces. The definition of the transpose may be seen to be independent of any bilinear form on the modules, unlike the adjoint ( below ). The continuous dual space of

Cracovian - Misplaced Pages Continue

1600-621: The bar notation is more common in pure mathematics . If a complex number is represented as a 2 × 2 {\displaystyle 2\times 2} matrix , the notations are identical, and the complex conjugate corresponds to the matrix transpose , which is a flip along the diagonal. The following properties apply for all complex numbers z {\displaystyle z} and w , {\displaystyle w,} unless stated otherwise, and can be proved by writing z {\displaystyle z} and w {\displaystyle w} in

1650-408: The columns contiguous) may improve performance by increasing memory locality . Ideally, one might hope to transpose a matrix with minimal additional storage. This leads to the problem of transposing an n  ×  m matrix in-place , with O(1) additional storage or at most storage much less than mn . For n  ≠  m , this involves a complicated permutation of the data elements that

1700-415: The columns of A are the rows of A , so the entry corresponds to the inner product of two rows of A . If p i j is the entry of the product, it is obtained from rows i and j in A . The entry p j i is also obtained from these rows, thus p i j = p j i , and the product matrix ( p i j ) is symmetric. Similarly, the product A A is a symmetric matrix. A quick proof of

1750-581: The conjugate of the conjugate of a complex number z {\displaystyle z} is z . {\displaystyle z.} In symbols, z ¯ ¯ = z . {\displaystyle {\overline {\overline {z}}}=z.} The product of a complex number with its conjugate is equal to the square of the number's modulus: z z ¯ = | z | 2 . {\displaystyle z{\overline {z}}={\left|z\right|}^{2}.} This allows easy computation of

1800-455: The element-by-element conjugation of A . {\displaystyle \mathbf {A} .} Contrast this to the property ( A B ) ∗ = B ∗ A ∗ , {\textstyle \left(\mathbf {AB} \right)^{*}=\mathbf {B} ^{*}\mathbf {A} ^{*},} where A ∗ {\textstyle \mathbf {A} ^{*}} represents

1850-774: The exponential function, and with the natural logarithm for nonzero arguments: z n ¯ = ( z ¯ ) n ,  for all  n ∈ Z {\displaystyle {\overline {z^{n}}}=\left({\overline {z}}\right)^{n},\quad {\text{ for all }}n\in \mathbb {Z} } exp ⁡ ( z ¯ ) = exp ⁡ ( z ) ¯ {\displaystyle \exp \left({\overline {z}}\right)={\overline {\exp(z)}}} ln ⁡ ( z ¯ ) = ln ⁡ ( z ) ¯  if  z  is not zero or

1900-451: The following methods: Formally, the i -th row, j -th column element of A is the j -th row, i -th column element of A : If A is an m × n matrix, then A is an n × m matrix. In the case of square matrices, A may also denote the T th power of the matrix A . For avoiding a possible confusion, many authors use left upperscripts, that is, they denote the transpose as A . An advantage of this notation

1950-1157: The form a + b i . {\displaystyle a+bi.} For any two complex numbers, conjugation is distributive over addition, subtraction, multiplication and division: z + w ¯ = z ¯ + w ¯ , z − w ¯ = z ¯ − w ¯ , z w ¯ = z ¯ w ¯ , and ( z w ) ¯ = z ¯ w ¯ , if  w ≠ 0. {\displaystyle {\begin{aligned}{\overline {z+w}}&={\overline {z}}+{\overline {w}},\\{\overline {z-w}}&={\overline {z}}-{\overline {w}},\\{\overline {zw}}&={\overline {z}}\;{\overline {w}},\quad {\text{and}}\\{\overline {\left({\frac {z}{w}}\right)}}&={\frac {\overline {z}}{\overline {w}}},\quad {\text{if }}w\neq 0.\end{aligned}}} A complex number

2000-460: The identity on C . {\displaystyle \mathbb {C} .} Thus the only two field automorphisms of C {\displaystyle \mathbb {C} } that leave the real numbers fixed are the identity map and complex conjugation. Once a complex number z = x + y i {\displaystyle z=x+yi} or z = r e i θ {\displaystyle z=re^{i\theta }}

2050-467: The inverse. Over a complex vector space, one often works with sesquilinear forms (conjugate-linear in one argument) instead of bilinear forms. The Hermitian adjoint of a map between such spaces is defined similarly, and the matrix of the Hermitian adjoint is given by the conjugate transpose matrix if the bases are orthonormal. Complex conjugate In mathematics , the complex conjugate of

Cracovian - Misplaced Pages Continue

2100-890: The line through 0 and u . {\displaystyle u.} These uses of the conjugate of z {\displaystyle z} as a variable are illustrated in Frank Morley 's book Inversive Geometry (1933), written with his son Frank Vigor Morley. The other planar real unital algebras, dual numbers , and split-complex numbers are also analyzed using complex conjugation. For matrices of complex numbers, A B ¯ = ( A ¯ ) ( B ¯ ) , {\textstyle {\overline {\mathbf {AB} }}=\left({\overline {\mathbf {A} }}\right)\left({\overline {\mathbf {B} }}\right),} where A ¯ {\textstyle {\overline {\mathbf {A} }}} represents

2150-424: The multiplication of planar real algebras is commutative , this reversal is not needed there. There is also an abstract notion of conjugation for vector spaces V {\textstyle V} over the complex numbers . In this context, any antilinear map φ : V → V {\textstyle \varphi :V\to V} that satisfies is called a complex conjugation , or

2200-422: The negation of its complex conjugate is called a skew-Hermitian matrix ; that is, A is skew-Hermitian if A square matrix whose transpose is equal to its inverse is called an orthogonal matrix ; that is, A is orthogonal if A square complex matrix whose transpose is equal to its conjugate inverse is called a unitary matrix ; that is, A is unitary if Let A and B be matrices and c be

2250-430: The products ( A ∧ B ) ∧ C and A ∧ ( B ∧ C ) will generally be different; thus, Cracovian multiplication is non- associative . Cracovians are an example of a quasigroup . Cracovians adopted a column-row convention for designating individual elements as opposed to the standard row-column convention of matrix analysis. This made manual multiplication easier, as one needed to follow two parallel columns (instead of

2300-428: The representation of some operation on linear maps. This leads to a much more general definition of the transpose that works on every linear map, even when linear maps cannot be represented by matrices (such as in the case of infinite dimensional vector spaces). In the finite dimensional case, the matrix representing the transpose of a linear map is the transpose of the matrix representing the linear map, independently of

2350-626: The row and column indices of the matrix A by producing another matrix, often denoted by A (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician Arthur Cayley . In the case of a logical matrix representing a binary relation R, the transpose corresponds to the converse relation R . The transpose of a matrix A , denoted by A , A , A , A ⊺ {\displaystyle A^{\intercal }} , A′ , A , A or A , may be constructed by any one of

2400-453: The symmetry of A A results from the fact that it is its own transpose: On a computer , one can often avoid explicitly transposing a matrix in memory by simply accessing the same data in a different order. For example, software libraries for linear algebra , such as BLAS , typically provide options to specify that certain matrices are to be interpreted in transposed order to avoid the necessity of data movement. However, there remain

2450-407: The term transpose to refer to the adjoint as defined here. The adjoint allows us to consider whether g  : Y → X is equal to u  : Y → X . In particular, this allows the orthogonal group over a vector space X with a quadratic form to be defined without reference to matrices (nor the components thereof) as the set of all linear maps X → X for which the adjoint equals

2500-441: The transpose u  : X → X i.e. B ( y , x ) = u (Ψ( y ))( x ) , we find that B ( x , y ) = B ( y , x ) . Here, Ψ is the natural homomorphism X → X into the double dual . If the vector spaces X and Y have respectively nondegenerate bilinear forms B X and B Y , a concept known as the adjoint , which is closely related to the transpose, may be defined: If u  : X → Y

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