Documentation

# sqrtm

Matrix square root

## Syntax

``X = sqrtm(A)``
``[X,residual] = sqrtm(A)``
``[X,alpha,condx] = sqrtm(A)``

## Description

example

````X = sqrtm(A)` returns the principal square root of the matrix `A`, that is, `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` might not have a square root. If exact singularity is detected, a warning is printed.```
````[X,residual] = sqrtm(A)` also returns 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)` returns stability factor `alpha` and an estimate of the matrix square root condition number of `X` in 1-norm, `condx`. The residual `norm(A-X^2,1)/norm(A,1)` is bounded approximately by `n*alpha*eps` and the 1-norm relative error in `X` is bounded approximately by `n*alpha*condx*eps`, where `n = max(size(A))`.```

## Examples

collapse all

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 6 ```

Calculate 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 2 ```

Consider a matrix that has four squareroots, `A`.

`$A=\left[\begin{array}{cc}7& 10\\ 15& 22\end{array}\right]$`

Two of the squareroots of `A` are given by `Y1` and `Y2`:

`${Y}_{1}=\left[\begin{array}{cc}1.5667& 1.7408\\ 2.6112& 4.1779\end{array}\right]$`

`${Y}_{2}=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right]$`

Confirm that `Y1` and `Y2` are squareroots of matrix `A`.

```A = [7 10; 15 22]; Y1 = [1.5667 1.7408; 2.6112 4.1779]; A - Y1*Y1```
```ans = 2×2 10-3 × -0.1258 -0.1997 -0.2995 -0.4254 ```
```Y2 = [1 2; 3 4]; A - Y2*Y2```
```ans = 2×2 0 0 0 0 ```

The other two squareroots 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 squareroots 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`:

`$S=\left[\begin{array}{cc}±0.3723& 0\\ 0& ±5.3723\end{array}\right]$`

Calculate the squareroot 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 4.1779 ```

## Input Arguments

collapse all

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, and `sqrtm` cannot be expected to produce one.

## Algorithms

The algorithm `sqrtm` uses is described in [3].

## References

[1] N.J. Higham, “Computing real square roots of a real matrix,” Linear Algebra and Appl., 88/89, pp. 405–430, 1987

[2] Bjorck, A. and S. Hammerling, “A Schur method for the square root of a matrix,” Linear Algebra 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,” Lecture Notes in Comput. Sci., 7782, Springer-Verlag, pp. 171–182, 2013