Stpm
Invertible matrix 1
Invertible matrix
In linear algebra an n-by-n (square) matrix A is called invertible or nonsingular or nondegenerate, if there exists an n-by-n matrix B such that where I n 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
[1]
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, 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, 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.
Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix
A.
Properties
Let A be a square n by n matrix over a field K (for example the field R of real numbers). Then the following statements are equivalent, but not necessarily true:
A is invertible.
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. det A ≠ 0. In general, a square matrix over a
Invertible matrix
In linear algebra an n-by-n (square) matrix A is called invertible or nonsingular or nondegenerate, if there exists an n-by-n matrix B such that where I n 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
[1]
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, 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, 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.
Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix
A.
Properties
Let A be a square n by n matrix over a field K (for example the field R of real numbers). Then the following statements are equivalent, but not necessarily true:
A is invertible.
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. det A ≠ 0. In general, a square matrix over a
References: • Cormen, Thomas H.; Leiserson, Charles E., Rivest, Ronald L., Stein, Clifford (2001) [1990]. "28.4: Inverting matrices" • Calculator for Singular or Non-Square Matrix Inverse (http://mjollnir.com/matrix/demo.html) • Derivative of inverse matrix (http://planetmath.org/?op=getobj&from=objects&id=6362) on