Invertible matrix

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In mathematics and especially linear algebra, an n-by-n (square) matrix A is called invertible, non-singular, or regular if there exists another n-by-n matrix B such that

AB = BA = In,

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. A square matrix that is not invertible is called 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 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 Rn×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 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.

Matrix inversion is the process of finding the matrix B that satisfies the prior equation for a given invertible matrix A.

Contents

Properties of invertible matrices

If 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 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 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 BA = In.
  • There is an n by n matrix B such that AB = In.
  • The transpose AT is an invertible matrix.
  • The matrix times its transpose, ATA, 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

(A−1)−1 = A

The inverse of an invertible matrix A multiplied by a scalar k yields the product of the inverse of both the matrix and the scalar

(kA)−1 = k−1A−1

The product of two invertible matrices A and B of the same size is again invertible, with the inverse given by

(AB)−1 = B−1A−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 A1, A2, ..., An 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 Ais 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 Ais. To remove An 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 An − 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 A1, 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 cofactors, 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}\right)^{T}={1 \over \begin{vmatrix}A\end{vmatrix}}
\begin{pmatrix}
C_{11} & C_{21} & \cdots & C_{j1} \\
C_{12} & \ddots &        & C_{j2} \\
\vdots &        & \ddots & \vdots \\
C_{1i} & \cdots & \cdots & C_{ji} \\
\end{pmatrix}

where |A| is the determinant of A, Cij is the matrix cofactor, and AT represents the matrix transpose.

In most practical applications, it is in fact not necessary to invert a matrix to solve a system of linear equations. This can instead be done using decomposition techniques like LU decomposition, which are much faster than inversion. Various fast algorithms for special classes of linear systems have also been developed.

Inversion of 2 x 2 matrices

The cofactor equation listed above yields the following result for 2 x 2 matrices. Inversion of these matrices can be done easily as follows:

A^{-1} = \begin{bmatrix}
a & b \\ c & d \\
\end{bmatrix}^{-1} =
\frac{1}{ad - bc} \begin{bmatrix}
d & -b \\ -c & a \\
\end{bmatrix}

Inversion of 3 x 3 matrices

The cofactor equation listed above yields the following result for 3 x 3 matrices. Inversion of these matrices can be done quite easily as follows:

A^{-1} = \begin{bmatrix}
a & b & c\\ d & e & f \\ g & h & i \\
\end{bmatrix}^{-1} =
\frac{1}{|A|} \begin{bmatrix}
ei - fh & ch - bi & bf - ce \\
fg - di & ai - cg & cd - af \\
dh - eg & bg - ah & ae - bd
\end{bmatrix}
| A | = a(eifh) − b(difg) + c(dheg)

Blockwise inversion

Matrices can also be inverted blockwisely by using the following analytic inversion formula:

\begin{bmatrix} A & B \\ C & D \end{bmatrix}^{-1} = \begin{bmatrix} A^{-1}+A^{-1}B(D-CA^{-1}B)^{-1}CA^{-1} & -A^{-1}B(D-CA^{-1}B)^{-1} \\ -(D-CA^{-1}B)^{-1}CA^{-1} & (D-CA^{-1}B)^{-1} \end{bmatrix}

where A, B, C and D are matrix sub-blocks of arbitrary size. This strategy is particularly advantageous if A is diagonal and (DCA − 1B) (the Schur complement of A) is a small matrix, since they are the only matrices requiring to be inverted.

This technique was invented by Volker Strassen, who also invented the Strassen algorithm for fast(er) matrix multiplication.

The Moore-Penrose pseudoinverse

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

See also


External links

de:Reguläre Matrix es:Matriz invertible fr:Matrice inversible it:Matrice invertibile he:מטריצה הפיכה nl:Inverse matrix ja:正則行列 pl:Macierz odwrotna ru:Обратная матрица zh:逆矩阵

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