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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. Sylvester ’s rank inequality : if A is an m × n matrix and B is n × k , then [ ii ] rank ( A ) + rank ( B ) − n ≤ rank ( A B ) . {\displaystyle ...
The transpose A T is an invertible matrix. A is row-equivalent to the n-by-n identity matrix I n. A is column-equivalent to the n-by-n identity matrix I n. A has n pivot positions. A has full rank: rank A = n. A has a trivial kernel: ker(A) = {0}. The linear transformation mapping x to Ax is bijective; that is, the equation Ax = b has exactly ...
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} .
The rank of a matrix is equal to the dimension of the row space, so row equivalent matrices must have the same rank. This is equal to the number of pivots in the reduced row echelon form. A matrix is invertible if and only if it is row equivalent to the identity matrix. Matrices A and B are row equivalent if and only if there exists an ...
A matrix (in this case the right-hand side of the Sherman–Morrison formula) is the inverse of a matrix (in this case +) if and only if = =. We first verify that the right hand side ( Y {\displaystyle Y} ) satisfies X Y = I {\displaystyle XY=I} .
In linear algebra, two rectangular m-by-n matrices A and B are called equivalent if = for some invertible n-by-n matrix P and some invertible m-by-m matrix Q.Equivalent matrices represent the same linear transformation V → W under two different choices of a pair of bases of V and W, with P and Q being the change of basis matrices in V and W respectively.
Decomposition: = 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 =.
Circular matrix or Coninvolutory matrix: A matrix whose inverse is equal to its entrywise complex conjugate: A −1 = A. Compare with unitary matrices. Congruent matrix: Two matrices A and B are congruent if there exists an invertible matrix P such that P T A P = B. Compare with similar matrices. EP matrix or Range-Hermitian matrix