sqrtm (MATLAB Function Reference)

Matrix square root

Y = sqrtm(X)
[Y,esterr] = sqrtm(X)

Description

Y = sqrtm(X)

is the matrix square root of X. Complex results are produced if X has negative eigenvalues. A warning message is printed if the computed Y*Y is not close to X.

[Y,esterr] = sqrtm(X)

does not print any warning message, but returns an estimate of the relative residual, norm(Y*Y-X)/norm(X).

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

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

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.

Finally, the matrix

 X =
     0    1
     0    0

does not have any square roots. There is no matrix Y, real or complex, for which Y*Y = X. The statement

Y = sqrtm(X)

produces several warning messages concerning accuracy and the answer

Y =

    1.0e+03 *

   0.0000+ 0.0000i   4.9354- 7.6863i
   0.0000+ 0.0000i   0.0000+ 0.0000i

Algorithm

The function sqrtm(X) is an abbreviation for funm(X,'sqrt'). The algorithm used by funm is based on a Schur decomposition. It can fail in certain situations where X has repeated eigenvalues. See funm for details.