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sqrtm - Matrix square root

Syntax

X = sqrtm(A) [X, resnorm] = sqrtm(A) [X, alpha, condest] = sqrtm(A)

Description

X = sqrtm(A) is the principal square root of the matrix A, i.e. 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 may not have a square root. A warning is printed if exact singularity is detected.

[X, resnorm] = sqrtm(A) does not print any warning, and returns the residual, norm(A-X^2,'fro')/norm(A,'fro').

[X, alpha, condest] = sqrtm(A) returns a stability factor alpha and an estimate condest of the matrix square root condition number of X. The residual norm(A-X^2,'fro')/norm(A,'fro') is bounded approximately by n*alpha*eps and the Frobenius norm relative error in X is bounded approximately by n*alpha*condest*eps, where n = max(size(A)).

Remarks

If X is real, symmetric and positive definite, or complex, Hermitian and positive definite, then so is the computed matrix square root.

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

Examples

Example 1

A matrix representation of the fourth difference operator is

X =
     5   -4    1    0    0
    -4    6   -4    1    0
     1   -4    6   -4    1
     0    1   -4    6   -4
     0    0    1   -4    5

This matrix is symmetric and positive definite. Its unique positive definite square root, Y = sqrtm(X), is a representation of the second difference operator.

 Y =
     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

Example 2

The matrix

X =
     7   10
    15   22

has four square roots. Two of them are

Y1 =
    1.5667    1.7408
    2.6112    4.1779

and

Y2 =
     1    2
     3    4

The other two are -Y1 and -Y2. All four can be obtained from the eigenvalues and vectors of X.

[V,D] = eig(X);
D =
    0.1386          0
         0    28.8614

The four square roots of the diagonal matrix D result from the four choices of sign in

S =
    ±0.3723          0
          0    ±5.3723

All four Ys are of the form

Y = V*S/V

The sqrtm function chooses the two plus signs and produces Y1, even though Y2 is more natural because its entries are integers.