sqrm: Square root of a quadratic matrix

Description

This function returns the square root of a quadratic and diagonalisable matrix.

Usage

sqrm(x, ...)

Arguments

matrix, must be quadratic.

...

The ellipsis argument is passed down to eigen().

Value

A matrix object and a scalar in case a $(1 \times 1)$ matrix has been provided.

Details

The computation of the square root of a matrix is based upon its eigen values and corresponding eigen vectors. The square matrix $A$ is diagonisable if there is a matrix $V$ such that $D = V^{-1}AV$, whereby $D$ is a diagonal matrix. This is only achieved if the eigen vectors of the $(n \times n)$ matrix $A$ constitute a basis of dimension $n$. The square root of $A$ is then $A^{1/2} = V D^{1/2} V'$.

Examples

Run this code

data(StockIndex)
S <- cov(StockIndex)
SR <- sqrm(S)
all.equal(crossprod(SR), S)