inv

Matrix inverse

Syntax

Y = inv(X)

Description

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

Inverse Matrix

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

Solve Linear System

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 = 
0.0160

Find the absolute and residual error of the calculation.

err_inv = norm(y-x)

err_inv = 
4.8457e-06

res_inv = norm(A*y-b)

res_inv = 
4.8952e-07

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

tic
z = A\b;
t1 = toc

t1 = 
0.0172

err_bs = norm(z-x)

err_bs = 
3.7705e-06

res_bs = norm(A*z-b)

res_bs = 
3.7080e-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

X — Input matrix

square matrix

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

More About

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)).

Extended Capabilities

C/C++ Code Generation

Generate C and C++ code using MATLAB® Coder™.

Usage notes and limitations:

• Singular matrix inputs can produce nonfinite values that differ from MATLAB^® results.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

• X must be nonsparse.

• The MATLAB inv function prints a warning if X is badly scaled or nearly singular. The gpuArray inv is unable to check for this condition. Take action to avoid this condition.

For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).

Version History

Introduced before R2006a

R2022a: Improved performance when inverting large triangular matrices

The inv function shows improved performance when operating on large triangular matrices.

For example, inverting a 5,000-by-5,000 upper triangular matrix is about 3.7x faster than in the previous release.

function timingInv
rng default
A = randn(5e3);
[~,R] = lu(A);

tic
Y = inv(R); 
toc
end

The approximate execution times are:

R2021b: 1.1 s

R2022a: 0.3 s

The code was timed on a Windows^® 10, Intel^® Xeon^® CPU W-2133 @ 3.60 GHz test system by calling the timingInv function.