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# sqrtm

Matrix square root

## Syntax

X = sqrtm(A)
[X, resnorm] = sqrtm(A)
[X, alpha, condest] = sqrtm(A)

## Description

X = sqrtm(A) is the principal square root of the matrix A, i.e. 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 may not have a square root. A warning is printed if exact singularity is detected.

[X, resnorm] = sqrtm(A) does not print any warning, and returns the residual, norm(A-X^2,'fro')/norm(A,'fro').

[X, alpha, condest] = sqrtm(A) returns a stability factor alpha and an estimate condest of the matrix square root condition number of X. The residual norm(A-X^2,'fro')/norm(A,'fro') is bounded approximately by n*alpha*eps and the Frobenius norm relative error in X is bounded approximately by n*alpha*condest*eps, where n = max(size(A)).

## Examples

### Example 1

A matrix representation of the fourth difference operator is

```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    5```

This matrix is symmetric and positive definite. Its unique positive definite square root, Y = sqrtm(A), 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 ```

### Example 2

The matrix

```A =
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 A.

```[V,D] = eig(A);
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.

## More About

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### Tips

If A is real, symmetric and positive definite, or complex, Hermitian and positive definite, then so is the computed matrix square root.

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.

## See Also

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