Like numbers, definition of

**square root** can be extended for matrices. A matrix

**B** is said to be a square root of

**A** if

**B**^{2}=A. For example square roots of the matrix

are

,

and their additive inverses.

There are a number of methods for calculating square roots of matrices. One of the frequently used method is diagonalization method. An

**n×n** matrix

**A** is diagonalizable if there is a matrix

**V** and a diagonal matrix

**D** such that

**A = VD**^{-1}V . This happens if and only if

**A** has

**n** eigenvectors which constitute a basis for

**C**^{n}. In this case,

**V** can be chosen to be the matrix with the

**n** eigenvectors as columns, and a square root of

**A** is

**VSV**^{-1}, where

**S** is any square root of

**D**. Indeed,

**(VD**^{1/2}V^{-1})^{2} = VD^{1/2}(V^{-1}V)D^{1/2}V^{-1} = VDV^{-1} = A .

As in the example, the matrix

can be diagonalized as

, where

and

.

**D** has principal square root

, giving the square root

.

Ref: Wikipedia