# Documentation

### This is machine translation

Translated by
Mouse over text to see original. Click the button below to return to the English verison of the page.

# sqrtm

Matrix square root

## Syntax

• X = sqrtm(A)
example
• [X,residual] = sqrtm(A)
• [X,alpha,condx] = sqrtm(A)

## Description

example

X = sqrtm(A) returns the principal square root of the matrix A, that is, X*X = A.X is the unique square root for which every eigenvalue has nonnegative real part. If A has any eigenvalues with negative real parts, then a complex result is produced. If A is singular, then A might not have a square root. If exact singularity is detected, a warning is printed.
[X,residual] = sqrtm(A) also returns the residual, residual = norm(A-X^2,1)/norm(A,1). This syntax does not print warnings if exact singularity is detected.
[X,alpha,condx] = sqrtm(A) returns stability factor alpha and an estimate of the matrix square root condition number of X in 1-norm, condx. The residual norm(A-X^2,1)/norm(A,1) is bounded approximately by n*alpha*eps and the 1-norm relative error in X is bounded approximately by n*alpha*condx*eps, where n = max(size(A)).

## Examples

collapse all

Create a matrix representation of the fourth difference operator, A. This matrix is symmetric and positive definite.

A = [5 -4 1 0 0; -4 6 -4 1 0; 1 -4 6 -4 1; 0 1 -4 6 -4; 0 0 1 -4 6] 
A = 5 -4 1 0 0 -4 6 -4 1 0 1 -4 6 -4 1 0 1 -4 6 -4 0 0 1 -4 6 

Calculate the unique positive definite square root of A using sqrtm. X is the matrix representation of the second difference operator.

X = round(sqrtm(A)) 
X = 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 

Consider a matrix that has four squareroots, A.

 

Two of the squareroots of A are given by Y1 and Y2:

 
 

Confirm that Y1 and Y2 are squareroots of matrix A.

A = [7 10; 15 22]; Y1 = [1.5667 1.7408; 2.6112 4.1779]; A - Y1*Y1 
ans = 1.0e-03 * -0.1258 -0.1997 -0.2995 -0.4254 
Y2 = [1 2; 3 4]; A - Y2*Y2 
ans = 0 0 0 0 

The other two squareroots of A are -Y1 and -Y2. All four of these roots can be obtained from the eigenvalues and eigenvectors of A. If [V,D] = eig(A), then the squareroots have the general form Y = V*S/V, where D = S*S and S has four choices of sign to produce four different values of Y:

 

Calculate the squareroot of A with sqrtm. The sqrtm function chooses the positive square roots and produces Y1, even though Y2 seems to be a more natural result.

Y = sqrtm(A) 
Y = 1.5667 1.7408 2.6112 4.1779 

## Input Arguments

collapse all

Input matrix, specified as a square matrix.

Data Types: single | double
Complex Number Support: Yes

collapse all

### Tips

• Some matrices, like A = [0 1; 0 0], do not have any square roots, real or complex, and sqrtm cannot be expected to produce one.

### Algorithms

The algorithm sqrtm uses is described in [3].

## References

[1] N.J. Higham, "Computing real square roots of a real matrix," Linear Algebra and Appl., 88/89, pp. 405–430, 1987

[2] Bjorck, A. and S. Hammerling, "A Schur method for the square root of a matrix," Linear Algebra and Appl., 52/53, pp. 127–140, 1983

[3] Deadman, E., Higham, N. J. and R. Ralha, "Blocked Schur algorithms for computing the matrix square root," Lecture Notes in Comput. Sci., 7782, Springer-Verlag, pp. 171–182, 2013