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The th column of an identity matrix is the unit vector, a vector whose th entry is 1 and 0 elsewhere. The determinant of the identity matrix is 1, and its trace is . The identity matrix is the only idempotent matrix with non-zero determinant. That is, it is the only matrix such that:
The rank of A is equal to r if and only if there exists an invertible m × m matrix X and an invertible n × n matrix Y such that = [], where I r denotes the r × r identity matrix and the three zero matrices have the sizes r × (n − r), (m − r) × r and (m − r) × (n − r).
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} .
Applicable to: m-by-n matrix A of rank r Decomposition: A = C F {\displaystyle A=CF} where C is an m -by- r full column rank matrix and F is an r -by- n full row rank matrix Comment: The rank factorization can be used to compute the Moore–Penrose pseudoinverse of A , [ 2 ] which one can apply to obtain all solutions of the linear system A x ...
The matrix determinant lemma performs a rank-1 update to a determinant. Woodbury matrix identity; Quasi-Newton method; Binomial inverse theorem; Bunch–Nielsen–Sorensen formula; Maxwell stress tensor contains an application of the Sherman–Morrison formula.
A common case is finding the inverse of a low-rank update A + UCV of A (where U only has a few columns and V only a few rows), or finding an approximation of the inverse of the matrix A + B where the matrix B can be approximated by a low-rank matrix UCV, for example using the singular value decomposition.
In mathematics, the general linear group of degree n is the set of n×n invertible matrices, together with the operation of ordinary matrix multiplication.This forms a group, because the product of two invertible matrices is again invertible, and the inverse of an invertible matrix is invertible, with the identity matrix as the identity element of the group.
Likewise, the Gram matrix of the rows or columns of a unitary matrix is the identity matrix. The rank of the Gram matrix of vectors in R k {\displaystyle \mathbb {R} ^{k}} or C k {\displaystyle \mathbb {C} ^{k}} equals the dimension of the space spanned by these vectors.