Y = sqrtm(X)
Y = sqrtm(X)
is one of the many matrices that satisfy
IfY
*Y = X
X
is symmetric and positive definite, then Y
is the unique positive definite square root.
This matrix is symmetric and positive definite. Its unique positive definite square root,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
Y = sqrtm(X)
, is a representation of the second difference operator.
The matrixY =
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
has four square roots. Two of them areX =
7 10
15 22
andY1 =
1.5667 1.7408
2.6112 4.1779
The other two areY2 =
1 2
3 4
-Y1
and -Y2
. All four can be obtained from the eigenvalues and vectors of X
.
The four square roots of the diagonal matrix[V,D] = eig(X);
D =
0.1386 0
0 28.8614
D
result from the four choices of sign in
All fourS =
±0.3723 0
0 ±5.3723
Y
s are of the form
TheY = V
*S/V
sqrtm
function chooses the two plus signs and produces Y1
, even though Y2
is more natural because its entries are integers.Finally, the matrix
does not have any square roots. There is no matrixX =
0 1
0 0
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
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.
expm
,funm
,logm
(c) Copyright 1994 by The MathWorks, Inc.