# matrix.sqrt: Matrix square root. In ExtremalDep: Extremal Dependence Models

## Description

Matrix square root using singular value decomposition (svd).

## Usage

 `1` ```matrix.sqrt(A) ```

## Arguments

 `A` A symmetric, positive definite matrix of size n x n.

## Details

Using svd, the matrix A can be written as \$\$ A = U S V^T, \$\$ where

• \$U\$ is called the left-singular vectors of \$A\$, which corresponds to the eigenvectors of \$AA^T\$,

• \$S\$ has entries \$S_i,j\$ which are the singular values of \$A\$,

• \$V\$ is called the right-singular vectors of \$A\$, which corresponds to the eigenvectors of \$A^T A\$.

The non-zero singular values of \$A\$ are the square roots of the non-zero eignevalues of both \$AA^T\$ and \$A^T A\$. Due to the symmetry and positive definiteness of \$A\$, without loss of generality we can set \$U=V\$. Thus \$A = V S V^T\$, and the columns of \$V\$ are eignevectors of \$A\$ with \$j-th\$ eignevalue begin \$s_j\$. Define \$\$ S^1/2 = diag((s_1^1/2,...,s_p^1/2)) \$\$ and hence we have \$\$ B = V S^1/2 V^T \$\$, a symmetric n x n matrix such that \$A = BB\$.

## Value

Return a n x n matrix.

## Author(s)

 ```1 2 3``` ``` A <- matrix(c(1,2,2,3),ncol=2,byrow=TRUE) print(A) B <- matrix.sqrt(A) ```