Definitions for invertible matrix
in·vert·ible ma·trix

WiktionaryRate this definition:5.0 / 1 vote

1. invertible matrixnoun

   a square matrix which, when multiplied by another (in either order), yields the identity matrix

WikipediaRate this definition:0.0 / 0 votes

1. Invertible matrix

   In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate), 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 In denotes 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−1. Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A. A square matrix that is not invertible is called singular or degenerate. A square matrix 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 finite 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 (m-by-n matrices for which m ≠ n) do not have an inverse. However, in some cases such 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 = In. If A has rank m (m ≤ n), then it has a right inverse, an n-by-m matrix B such that AB = Im. While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any ring. However, in the case of the 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 being nonzero. For a noncommutative ring, 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 GLn(R).

WikidataRate this definition:2.0 / 2 votes

1. Invertible matrix

   In linear algebra an n-by-n matrix A is called invertible if there exists an n-by-n matrix B such that where In denotes 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 inverse of A, denoted by A−1. It follows from the theory of matrices that if for finite square matrices A and B, then also Non-square matrices do not have an inverse. However, in some cases such 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, then A has a left inverse: an n-by-m matrix B such that BA = I. If A has rank m, then it has a right inverse: an n-by-m matrix B such that AB = I. A square matrix that is not invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is 0. Singular matrices are rare in the sense that if you pick a random square matrix over a continuous uniform distribution on its entries, it will almost surely not be singular. While the most common case is that of matrices over the real or complex numbers, all these definitions can be given for matrices over any commutative ring. However, in this case the condition for a square matrix to be invertible is that its determinant is invertible in the ring, which in general is a much stricter requirement than being nonzero. The conditions for existence of left-inverse resp. right-inverse are more complicated since a notion of rank does not exist over rings.

How to pronounce invertible matrix?

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Numerology

1. Chaldean Numerology

   The numerical value of invertible matrix in Chaldean Numerology is: 6

2. Pythagorean Numerology

   The numerical value of invertible matrix in Pythagorean Numerology is: 3

References

1. ^ Wiktionary
   https://en.wiktionary.org/wiki/Invertible_Matrix
2. ^ Wikipedia
   https://en.wikipedia.org/wiki/Invertible_Matrix
3. ^ Wikidata
   https://www.wikidata.org/w/index.php?search=invertible matrix