Matrix square root
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Syntax
X = sqrtm(A)
[X,residual] = sqrtm(A)
[X,alpha,condx] = sqrtm(A)
Description
example
X = sqrtm(A) returnsthe principal square root of the matrix A, thatis, X*X = A.
X is the unique square root for which everyeigenvalue has nonnegative real part. If A hasany eigenvalues with negative real parts, then a complex result isproduced. If A is singular, then A mightnot have a square root. If exact singularity is detected, a warningis printed.
[X,residual] = sqrtm(A) alsoreturns 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) returnsstability factor alpha and an estimate of the matrixsquare root condition number of X in 1-norm, condx.The residual norm(A-X^2,1)/norm(A,1) is boundedapproximately by n*alpha*eps and the 1-norm relativeerror in X is bounded approximately by n*alpha*condx*eps,where n=max(size(A)).
Examples
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Matrix Square Root of Difference Operator
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 = 5×5 5 -4 1 0 0 -4 6 -4 1 0 1 -4 6 -4 1 0 1 -4 6 -4 0 0 1 -4 6Calculate 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 = 5×5 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2 -1 0 0 0 -1 2Matrix with Multiple Square Roots
Open Live Script
Consider a matrix that has four square roots, A.
Two of the square roots of A are given by Y1 and Y2:
Confirm that Y1 and Y2 are square roots of matrix A.
A = [7 10; 15 22];Y1 = [1.5667 1.7408; 2.6112 4.1779];A - Y1*Y1
ans = 2×210-3 × -0.1258 -0.1997 -0.2995 -0.4254Y2 = [1 2; 3 4];A - Y2*Y2
ans = 2×2 0 0 0 0The other two square roots 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 square roots 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 square root 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 = 2×2 1.5667 1.7408 2.6112 4.1779Input Arguments
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A — Input matrix
square matrix
Input matrix, specified as a square matrix.
Data Types: single | double
Complex Number Support: Yes
Tips
Some matrices, like
A = [0 1; 0 0],do not have any square roots, real or complex, andsqrtmcannotbe expected to produce one.
Algorithms
The algorithm sqrtm uses is described in [3].
References
[1] N.J. Higham, “Computing real squareroots of a real matrix,” Linear Algebra and Appl.,88/89, pp. 405–430, 1987
[2] Bjorck, A. and S. Hammerling, “ASchur method for the square root of a matrix,” LinearAlgebra 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,” LectureNotes in Comput. Sci., 7782, Springer-Verlag, pp. 171–182,2013
Extended Capabilities
C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.
Usage notes and limitations:
When the input matrix contains a nonfinite value, the generated code does not issue an error. Instead, the output contains
NaNvalues.In the generated code, the output of
sqrtmis always complex.Code generation does not support sparse matrix inputs for this function.
Thread-Based Environment
Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.
This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.
Version History
Introduced before R2006a
See Also
expm | funm | logm
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