Invertible Matrix

AB = BA = I_n \

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

:

AB = I \

for square matrices

A

and

B

, then also

:

BA = I \

.

A square matrix that is not invertible is called singular or '''degenerate'''. 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 .

As a rule of thumb, almost all matrices are invertible. Over the field of real numbers, this can be made precise as follows: the set of singular ''n''-by-''n'' matrices, considered as a subset of

R^{n 	imes n}

, is a Null Set , i.e., has Lebesgue Measure Zero . Intuitively, this means that if you pick a random square matrix over the reals, the Probability that it will be singular is zero. This is true because singular matrices can be thought of as the roots of the polynomial function given by the determinant. In practice however, one may encounter non-invertible matrices. And in Numerical Calculations , matrices which are invertible, but close to a non-invertible matrix, can still be problematic.

''Matrix inversion'' is the process of finding the matrix

B

that satisfies the prior equation for a given invertible matrix

A

.

PROPERTIES OF INVERTIBLE MATRICES

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:

$A$ is invertible.

$A$ is Row-equivalent to the ''n''-by-''n'' Identity Matrix $I_n$ .

$A$ has ''n'' pivot positions.

Det $A$ ≠ 0.

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 $K^n$ .

The columns of $A$ are Linearly Independent .

The columns of $A$ span $K^n$ (i.e. Col $A = K^n$ ).

The columns of $A$ form a Basis of $K^n$ .

The linear transformation mapping $x$ to $Ax$ is a Bijection from $K^n$ to $K^n$ .

There is an ''n'' by ''n'' matrix $B$ such that $AB = I_n$ .

The Transpose $A^T$ is an invertible matrix.

The matrix times its transpose, $A^T imes A$ is an invertible matrix.

The number 0 is not an Eigenvalue of $A$ .

In general, a square matrix over a Commutative Ring is invertible if and only if its Determinant is a Unit in that ring.

The inverse of an invertible matrix

A

is itself invertible, with
:

\left(A^{-1}
ight)^{-1} = A

.
The inverse of an invertible matrix

K

multiplied by a scalar

k

yields the product of the inverse of both the matrix and the scalar
:

\left(kA
ight)^{-1} = k^{-1}A^{-1}

.
The product of two invertible matrices

A

and

B

of the same size is again invertible, with the inverse given by
:

\left(AB
ight)^{-1} = B^{-1}A^{-1}

(note that the order of the factors is reversed.) As a consequence, the set of invertible ''n''-by-''n'' matrices forms a Group , known as the General Linear Group Gl(''n'').

Proof for matrix product rule

If

A_1

A_2

, ...,

A_n

are nonsingular square matrices over a field, then

:

(A_1A_2\cdots A_n)^{-1} = A_n^{-1}A_{n-1}^{-1}\cdots A_1^{-1}

It becomes evident why this is the case if one attempts to find an inverse for the product of the

A_i

s from first principles, that is, that we wish to determine

B

such that
:

(A_1A_2\cdots A_n)B=I

where

B

is some matrix, in terms of the

A_i

s.
To remove

A_n

from the product, we can then write
:

(A_1A_2\cdots A_n)A_n^{-1}B'=I

where

B'

is some matrix, which would reduce the equation to
:

(A_1A_2\cdots A_{n-1})B'=I

Likewise, then, from
:

(A_1A_2\cdots A_n)A_n^{-1}B'=I

we use the same technique, removing

A_{n-1}

from the equation, yielding
:

(A_1A_2\cdots A_{n-1}A_n)A_n^{-1}A_{n-1}^{-1}B''=I

where

B'

is some matrix, which, when simplified, gives
:

(A_1A_2\cdots A_{n-2})B''=I

If one repeat the process up to

A_1

, the above property is established.

METHODS OF MATRIX INVERSION

Gauss-Jordan elimination

Gauss-Jordan Elimination is an Algorithm that can be used to determine whether a given matrix is invertible and to find the inverse. An alternative is the LU Decomposition which generates an upper and a lower triangular matrices which are easier to invert. For special purposes, it may be convenient to invert matrices by treating ''mn''-by-''mn'' matrices as ''m''-by-''m'' matrices of ''n''-by-''n'' matrices, and applying one or another formula recursively (other sized matrices can be padded out with dummy rows and columns). For other purposes, a variant of Newton's Method may be convenient (particularly when dealing with families of related matrices, so inverses of earlier matrices can be used to seed generating inverses of later matrices).

Analytic solution

Writing another special matrix of Cofactor s, known as an Adjugate matrix, can also be an efficient way to calculate the inverse of ''small'' matrices (since this method is essentially recursive, it becomes inefficient for large matrices). To determine the inverse, we calculate a matrix of cofactors:

:

A^{-1}={1 \over \begin{vmatrix}A\end{vmatrix}}\left(C_{ij}
ight)^{T}={1 \over \begin{vmatrix}A\end{vmatrix}}
\begin{pmatrix}
C_{11} & C_{21} & \cdots & C_{j1} \
C_{12} & \ddots &        & C_{j2} \
dots &        & \ddots & dots \
C_{1i} & \cdots & \cdots & C_{ji} \
\end{pmatrix}

:<math>A a(ei-fh) - b(di-fg) + c(dh-eg) \ </math>

This formula can be found by differentiating the identity
:

A^{-1}A = I

.

THE MOORE-PENROSE PSEUDOINVERSE

Some of the properties of inverse matrices are shared by (Moore-Penrose) Pseudoinverse s, which can be defined for any ''m''-by-''n'' matrix.

SEE ALSO

Matrix Decomposition

Matrix Multiplication

Pseudoinverse (Moore-Penrose inverse)

Singular Value Decomposition

REFERENCES

Thomas H. Cormen , Charles E. Leiserson , Ronald L. Rivest , and Clifford Stein . '' Introduction To Algorithms '', Second Edition. MIT Press and McGraw-Hill, 2001. ISBN 0262032937. Section 28.4: Inverting matrices, pp.755–760.

EXTERNAL LINKS

Inverse Matrix Calculator

Moore Penrose Pseudoinverse

	CATEGORIES ABOUT INVERTIBLE MATRIX

	linear algebra

	matrices

	determinants

	matrix theory

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Invertible Matrix