Documentation

inv

Matrix inverse

Description

example

Y = inv(X) computes the inverse of square matrix X.

• X^(-1) is equivalent to inv(X).

• x = A\b is computed differently than x = inv(A)*b and is recommended for solving systems of linear equations.

Examples

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Compute the inverse of a 3-by-3 matrix.

X = [1 0 2; -1 5 0; 0 3 -9]
X = 3×3

1     0     2
-1     5     0
0     3    -9

Y = inv(X)
Y = 3×3

0.8824   -0.1176    0.1961
0.1765    0.1765    0.0392
0.0588    0.0588   -0.0980

Check the results. Ideally, Y*X produces the identity matrix. Since inv performs the matrix inversion using floating-point computations, in practice Y*X is close to, but not exactly equal to, the identity matrix eye(size(X)).

Y*X
ans = 3×3

1.0000    0.0000   -0.0000
0    1.0000   -0.0000
0   -0.0000    1.0000

Examine why solving a linear system by inverting the matrix using inv(A)*b is inferior to solving it directly using the backslash operator, x = A\b.

Create a random matrix A of order 500 that is constructed so that its condition number, cond(A), is 1e10, and its norm, norm(A), is 1. The exact solution x is a random vector of length 500, and the right side is b = A*x. Thus the system of linear equations is badly conditioned, but consistent.

n = 500;
Q = orth(randn(n,n));
d = logspace(0,-10,n);
A = Q*diag(d)*Q';
x = randn(n,1);
b = A*x;

Solve the linear system A*x = b by inverting the coefficient matrix A. Use tic and toc to get timing information.

tic
y = inv(A)*b;
t = toc
t = 0.0185

Find the absolute and residual error of the calculation.

err_inv = norm(y-x)
err_inv = 3.6334e-06
res_inv = norm(A*y-b)
res_inv = 4.4299e-07

Now, solve the same linear system using the backslash operator \.

tic
z = A\b;
t1 = toc
t1 = 0.0186
err_bs = norm(z-x)
err_bs = 2.1746e-06
res_bs = norm(A*z-b)
res_bs = 2.4410e-15

The backslash calculation is quicker and has less residual error by several orders of magnitude. The fact that err_inv and err_bs are both on the order of 1e-6 simply reflects the condition number of the matrix.

The behavior of this example is typical. Using A\b instead of inv(A)*b is two to three times faster, and produces residuals on the order of machine accuracy relative to the magnitude of the data.

Input Arguments

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Input matrix, specified as a square matrix. If X is badly scaled or nearly singular, then the inv calculation loses numerical accuracy. Use rcond or cond to check the condition number of the matrix.

Data Types: single | double
Complex Number Support: Yes

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Matrix Inverse

A matrix X is invertible if there exists a matrix Y of the same size such that $XY=YX={I}_{n}$, where ${I}_{n}$ is the n-by-n identity matrix. The matrix Y is called the inverse of X.

A matrix that has no inverse is singular. A square matrix is singular only when its determinant is exactly zero.

Tips

• It is seldom necessary to form the explicit inverse of a matrix. A frequent misuse of inv arises when solving the system of linear equations Ax = b. One way to solve the equation is with x = inv(A)*b. A better way, from the standpoint of both execution time and numerical accuracy, is to use the matrix backslash operator x = A\b. This produces the solution using Gaussian elimination, without explicitly forming the inverse. See mldivide for further information.

Algorithms

inv performs an LU decomposition of the input matrix (or an LDL decomposition if the input matrix is Hermitian). It then uses the results to form a linear system whose solution is the matrix inverse inv(X). For sparse inputs, inv(X) creates a sparse identity matrix and uses backslash, X\speye(size(X)).