sqrtm

Purpose

Matrix square root.

Synopsis

Y = sqrtm(X)

Description

Y = sqrtm(X) is one of the many matrices that satisfy

Y*Y = X
If X is symmetric and positive definite, then Y is the unique positive definite square root.

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

WARNING: Result from FUNM is probably inaccurate.
           
Y =
     0    0
     0    0

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.

See Also

expm, funm, logm

(c) Copyright 1994 by The MathWorks, Inc.