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36-513: [REDACTED] Look up singular in Wiktionary, the free dictionary. Singular may refer to: Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms Singular or sounder, a group of boar, see List of animal names Singular (band) , a Thai jazz pop duo Singular: Act I , a 2018 studio album by Sabrina Carpenter Singular: Act II ,

72-510: A field K (e.g., the field ⁠ R {\displaystyle \mathbb {R} } ⁠ of real numbers). The following statements are equivalent, i.e., they are either all true or all false for any given matrix: Furthermore, the following properties hold for an invertible matrix A : The rows of the inverse matrix V of a matrix U are orthonormal to the columns of U (and vice versa interchanging rows for columns). To see this, suppose that UV = VU = I where

108-430: A 2019 studio album by Sabrina Carpenter Mathematics [ edit ] Singular homology SINGULAR , an open source Computer Algebra System (CAS) Singular matrix , a matrix that is not invertible Singular measure , a measure or probability distribution whose support has zero Lebesgue (or other) measure Singular cardinal, an infinite cardinal number that is not a regular cardinal Singular point of

144-483: A curve , in geometry See also [ edit ] Singularity (disambiguation) Singulair , Merck trademark for the drug Montelukast Cingular Wireless , a mobile network operator in North America Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Singular . If an internal link led you here, you may wish to change

180-409: A matrix may have a left inverse or right inverse . If A is m -by- n and the rank of A is equal to n , ( n ≤ m ), then A has a left inverse, an n -by- m matrix B such that BA = I n . If A has rank m ( m ≤ n ), then it has a right inverse, an n -by- m matrix B such that AB = I m . While the most common case is that of matrices over

216-418: A matrix that is not invertible Singular measure , a measure or probability distribution whose support has zero Lebesgue (or other) measure Singular cardinal, an infinite cardinal number that is not a regular cardinal Singular point of a curve , in geometry See also [ edit ] Singularity (disambiguation) Singulair , Merck trademark for the drug Montelukast Cingular Wireless ,

252-480: A mobile network operator in North America Topics referred to by the same term [REDACTED] This disambiguation page lists articles associated with the title Singular . If an internal link led you here, you may wish to change the link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Singular&oldid=1250591878 " Category : Disambiguation pages Hidden categories: Short description

288-556: A square matrix that is not invertible is called singular or degenerate . A square matrix with entries in a field is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any bounded region on the number line or complex plane , the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices, i.e. m -by- n matrices for which m ≠ n , do not have an inverse. However, in some cases such

324-402: Is a null set , that is, has Lebesgue measure zero. This is true because singular matrices are the roots of the determinant function. This is a continuous function because it is a polynomial in the entries of the matrix. Thus in the language of measure theory , almost all n -by- n matrices are invertible. Furthermore, the set of n -by- n invertible matrices is open and dense in

360-421: Is a non-invertible matrix We can see the rank of this 2-by-2 matrix is 1, which is n − 1 ≠ n , so it is non-invertible. Consider the following 2-by-2 matrix: The matrix B {\displaystyle \mathbf {B} } is invertible. To check this, one can compute that det B = − 1 2 {\textstyle \det \mathbf {B} =-{\frac {1}{2}}} , which

396-519: Is called an involutory matrix . The adjugate of a matrix A can be used to find the inverse of A as follows: If A is an invertible matrix, then It follows from the associativity of matrix multiplication that if for finite square matrices A and B , then also Over the field of real numbers, the set of singular n -by- n matrices, considered as a subset of ⁠ R n × n , {\displaystyle \mathbb {R} ^{n\times n},} ⁠

SECTION 10

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432-400: Is convenient to find a suitable starting seed: Victor Pan and John Reif have done work that includes ways of generating a starting seed. Newton's method is particularly useful when dealing with families of related matrices that behave enough like the sequence manufactured for the homotopy above: sometimes a good starting point for refining an approximation for the new inverse can be

468-476: Is different from Wikidata All article disambiguation pages All disambiguation pages Singular matrix An n -by- n square matrix A is called invertible (also nonsingular , nondegenerate or rarely regular ) if there exists an n -by- n square matrix B such that A B = B A = I n , {\displaystyle \mathbf {AB} =\mathbf {BA} =\mathbf {I} _{n},} where I n denotes

504-594: Is first created with the left side being the matrix to invert and the right side being the identity matrix . Then, Gaussian elimination is used to convert the left side into the identity matrix, which causes the right side to become the inverse of the input matrix. For example, take the following matrix: A = ( − 1 3 2 1 − 1 ) . {\displaystyle \mathbf {A} ={\begin{pmatrix}-1&{\tfrac {3}{2}}\\1&-1\end{pmatrix}}.} The first step to compute its inverse

540-445: Is invertible and its inverse is given by where Q is the square ( N × N ) matrix whose i th column is the eigenvector q i {\displaystyle q_{i}} of A , and Λ is the diagonal matrix whose diagonal entries are the corresponding eigenvalues, that is, Λ i i = λ i . {\displaystyle \Lambda _{ii}=\lambda _{i}.} If A

576-416: Is non-zero. As an example of a non-invertible, or singular, matrix, consider the matrix The determinant of C {\displaystyle \mathbf {C} } is 0, which is a necessary and sufficient condition for a matrix to be non-invertible. Gaussian elimination is a useful and easy way to compute the inverse of a matrix. To compute a matrix inverse using this method, an augmented matrix

612-403: Is symmetric, Q is guaranteed to be an orthogonal matrix , therefore Q − 1 = Q T . {\displaystyle \mathbf {Q} ^{-1}=\mathbf {Q} ^{\mathrm {T} }.} Furthermore, because Λ is a diagonal matrix, its inverse is easy to calculate: If matrix A is positive definite , then its inverse can be obtained as where L

648-1030: Is that the process of Gaussian elimination can be viewed as a sequence of applying left matrix multiplication using elementary row operations using elementary matrices ( E n {\displaystyle \mathbf {E} _{n}} ), such as E n E n − 1 ⋯ E 2 E 1 A = I . {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {A} =\mathbf {I} .} Applying right-multiplication using A − 1 , {\displaystyle \mathbf {A} ^{-1},} we get E n E n − 1 ⋯ E 2 E 1 I = I A − 1 . {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} =\mathbf {I} \mathbf {A} ^{-1}.} And

684-515: Is the determinant of A , C is the matrix of cofactors, and C represents the matrix transpose . The cofactor equation listed above yields the following result for 2 × 2 matrices. Inversion of these matrices can be done as follows: This is possible because 1/( ad − bc ) is the reciprocal of the determinant of the matrix in question, and the same strategy could be used for other matrix sizes. The Cayley–Hamilton method gives A computationally efficient 3 × 3 matrix inversion

720-437: Is the lower triangular Cholesky decomposition of A , and L * denotes the conjugate transpose of L . Writing the transpose of the matrix of cofactors , known as an adjugate matrix , can also be an efficient way to calculate the inverse of small matrices, but this recursive method is inefficient for large matrices. To determine the inverse, we calculate a matrix of cofactors: so that where | A |

756-430: Is to create the augmented matrix ( − 1 3 2 1 0 1 − 1 0 1 ) . {\displaystyle \left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\1&-1&0&1\end{array}}\right).} Call the first row of this matrix R 1 {\displaystyle R_{1}} and

SECTION 20

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792-399: The n -by- n identity matrix and the multiplication used is ordinary matrix multiplication . If this is the case, then the matrix B is uniquely determined by A , and is called the (multiplicative) inverse of A , denoted by A . Matrix inversion is the process of finding the matrix which when multiplied by the original matrix gives the identity matrix. Over a field ,

828-422: The real or complex numbers, all these definitions can be given for matrices over any algebraic structure equipped with addition and multiplication (i.e. rings ). However, in the case of a ring being commutative , the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a stricter requirement than it being nonzero. For a noncommutative ring ,

864-441: The topological space of all n -by- n matrices. Equivalently, the set of singular matrices is closed and nowhere dense in the space of n -by- n matrices. In practice however, one may encounter non-invertible matrices. And in numerical calculations , matrices which are invertible, but close to a non-invertible matrix, can still be problematic; such matrices are said to be ill-conditioned . An example with rank of n − 1

900-554: The Gauss–Jordan algorithm which has been contaminated by small errors due to imperfect computer arithmetic . The Cayley–Hamilton theorem allows the inverse of A to be expressed in terms of det( A ) , traces and powers of A : where n is size of A , and tr( A ) is the trace of matrix A given by the sum of the main diagonal . The sum is taken over s and the sets of all k l ≥ 0 {\displaystyle k_{l}\geq 0} satisfying

936-409: The already obtained inverse of a previous matrix that nearly matches the current matrix, for example, the pair of sequences of inverse matrices used in obtaining matrix square roots by Denman–Beavers iteration ; this may need more than one pass of the iteration at each new matrix, if they are not close enough together for just one to be enough. Newton's method is also useful for "touch up" corrections to

972-419: The augumented matrix by combining A with I and applying Gaussian elimination . The two portions will be transformed using the same sequence of elementary row operations. When the left portion becomes I , the right portion applied the same elementary row operation sequence will become A . A generalization of Newton's method as used for a multiplicative inverse algorithm may be convenient, if it

1008-542: The free dictionary. Singular may refer to: Singular, the grammatical number that denotes a unit quantity, as opposed to the plural and other forms Singular or sounder, a group of boar, see List of animal names Singular (band) , a Thai jazz pop duo Singular: Act I , a 2018 studio album by Sabrina Carpenter Singular: Act II , a 2019 studio album by Sabrina Carpenter Mathematics [ edit ] Singular homology SINGULAR , an open source Computer Algebra System (CAS) Singular matrix ,

1044-553: The identity matrix on the left side and the inverse matrix on the right: ( 1 0 2 3 0 1 2 2 ) . {\displaystyle \left({\begin{array}{cc|cc}1&0&2&3\\0&1&2&2\end{array}}\right).} Thus, A − 1 = ( 2 3 2 2 ) . {\displaystyle \mathbf {A} ^{-1}={\begin{pmatrix}2&3\\2&2\end{pmatrix}}.} The reason it works

1080-406: The inverse of a square matrix in some instances, where a set of orthogonal vectors (but not necessarily orthonormal vectors) to the columns of U are known. In which case, one can apply the iterative Gram–Schmidt process to this initial set to determine the rows of the inverse V . A matrix that is its own inverse (i.e., a matrix A such that A = A , and consequently A = I ),

1116-495: The linear Diophantine equation The formula can be rewritten in terms of complete Bell polynomials of arguments t l = − ( l − 1 ) ! tr ⁡ ( A l ) {\displaystyle t_{l}=-(l-1)!\operatorname {tr} \left(A^{l}\right)} as This is described in more detail under Cayley–Hamilton method . If matrix A can be eigendecomposed, and if none of its eigenvalues are zero, then A

Singular - Misplaced Pages Continue

1152-451: The link to point directly to the intended article. Retrieved from " https://en.wikipedia.org/w/index.php?title=Singular&oldid=1250591878 " Category : Disambiguation pages Hidden categories: Short description is different from Wikidata All article disambiguation pages All disambiguation pages singular [REDACTED] Look up singular in Wiktionary,

1188-504: The right side I A − 1 = A − 1 , {\displaystyle \mathbf {I} \mathbf {A} ^{-1}=\mathbf {A} ^{-1},} which is the inverse we want. To obtain E n E n − 1 ⋯ E 2 E 1 I , {\displaystyle \mathbf {E} _{n}\mathbf {E} _{n-1}\cdots \mathbf {E} _{2}\mathbf {E} _{1}\mathbf {I} ,} we create

1224-665: The rows of V are denoted as v i T {\displaystyle v_{i}^{\mathrm {T} }} and the columns of U as u j {\displaystyle u_{j}} for 1 ≤ i , j ≤ n . {\displaystyle 1\leq i,j\leq n.} Then clearly, the Euclidean inner product of any two v i T u j = δ i , j . {\displaystyle v_{i}^{\mathrm {T} }u_{j}=\delta _{i,j}.} This property can also be useful in constructing

1260-1435: The second row R 2 {\displaystyle R_{2}} . Then, add row 1 to row 2 ( R 1 + R 2 → R 2 ) . {\displaystyle (R_{1}+R_{2}\to R_{2}).} This yields ( − 1 3 2 1 0 0 1 2 1 1 ) . {\displaystyle \left({\begin{array}{cc|cc}-1&{\tfrac {3}{2}}&1&0\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).} Next, subtract row 2, multiplied by 3, from row 1 ( R 1 − 3 R 2 → R 1 ) , {\displaystyle (R_{1}-3\,R_{2}\to R_{1}),} which yields ( − 1 0 − 2 − 3 0 1 2 1 1 ) . {\displaystyle \left({\begin{array}{cc|cc}-1&0&-2&-3\\0&{\tfrac {1}{2}}&1&1\end{array}}\right).} Finally, multiply row 1 by −1 ( − R 1 → R 1 ) {\displaystyle (-R_{1}\to R_{1})} and row 2 by 2 ( 2 R 2 → R 2 ) . {\displaystyle (2\,R_{2}\to R_{2}).} This yields

1296-428: The usual determinant is not defined. The conditions for existence of left-inverse or right-inverse are more complicated, since a notion of rank does not exist over rings. The set of n × n invertible matrices together with the operation of matrix multiplication and entries from ring R form a group , the general linear group of degree n , denoted GL n ( R ) . Let A be a square n -by- n matrix over

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