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In linear algebra an n-by-n (square) matrix A is called invertible (some authors use nonsingular or nondegenerate) 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

[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 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.

Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.Contents [hide] 1 Properties

1.1 Density

2 Methods of matrix inversion

2.1 Gaussian elimination

2.2 Eigendecomposition

2.3 Cholesky decomposition

2.4 Analytic solution

2.4.1 Inversion of 2×2 matrices

2.4.2 Inversion of 3×3 matrices

2.5 Blockwise inversion

2.6 By Neumann series

3 Derivative of the matrix inverse

4 Moore–Penrose pseudoinverse

5 Applications

5.1 Matrix inverses in real-time simulations

5.2 Matrix inverses in MIMO wireless communication

6 See also

7 Notes

8 References

9 External links

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Properties

Let A be a square n by n matrix over a field K (for example the field R of real numbers). The following statements are equivalent: A is invertible.

A is row-equivalent to the n-by-n identity matrix In.

A is column-equivalent to the n-by-n identity matrix In.

A has n pivot positions.

det A ≠ 0. In general, a square matrix over a commutative ring is invertible if and only if its determinant is a unit in that ring. rank A = n.

The equation Ax = 0 has only the trivial solution x = 0 (i.e., Null A = {0}) The equation Ax = b has exactly one solution for each b in Kn. The columns of A are linearly independent.

The columns of A span Kn (i.e. Col A = Kn).

The columns of A form a basis of Kn.

The linear transformation mapping x to Ax is a bijection from Kn to Kn. There is an n by n matrix B such that AB = In = BA.

The transpose AT is an invertible matrix (hence rows of A are linearly independent, span Kn, and form a basis of Kn). The number 0 is not an eigenvalue of A.

The matrix A can be expressed as a finite product of elementary matrices.

Furthermore, the following properties hold for an invertible matrix A: (A−1)−1 = A;

(kA)−1 = k−1A−1 for nonzero scalar k;

(AT)−1 = (A−1)T;

For any invertible n-by-n matrices A and B, (AB)−1 = B−1A−1. More generally, if A1,...,Ak are invertible n-by-n matrices, then (A1A2⋯Ak−1Ak)−1 = Ak−1Ak−1−1⋯A2−1A1−1; det(A−1) = det(A)−1.

A matrix that is its own inverse, i.e. A = A-1 and A2 = I, is called an involution. [edit]

Density

Over the field of real numbers, the set of...