How do you find the Square Root of a Matrix????

Question

Find a Square root of the Matrix

 

A =   [4  3]

        [2  3] 

 

This is a 2x2 matrix

 

 

 

Answer 1

The eigenvalues of A are 1 and 6

 

A basis for the eigenspace corresponding to the eigenvalue 1 is <-1,1>

 

A basis for the eigenspace corresponding to the eigenvalue 6 is <3,2>

 

Let P be the 2x2 matrix ⌈-1    3⌉

                                          ⌊ 1     2⌋

 

Then P^-1AP = D = diagonal 2x2 matrix with 1 and 6 on the main                                     diagonal

 

So, A = PDP^-1 

 

Now, let D₁ = 2x2 diagonal matrix with √1 and√6 on the main diagonal.  Note that (D₁)² = D.

 

Then, (PD₁P^-1)(PD₁P^-1) = P(D₁)²P^-1 = PDP^-1 = A

 

So, PD₁P^-1 is the square root of A.