Ex:            The following Matlab® code shows how to diagonalize a matrix A:

where

S has eigenvectors as its columns

Λ is diagonal with eigenvalues on its diagonal

 

 

syms a b c d

syms A S D

 

A = [a, b; c, d]

A =

[ a, b]

[ c, d]

 

[V,D] = eig(A)

V =

[ -(-1/2*a+1/2*d-1/2*(a^2-2*a*d+d^2+4*b*c)^(1/2))/c, -(-1/2*a+1/2*d+1/2*(a^2-2*a*d+d^2+4*b*c)^(1/2))/c]

[ 1, 1]

D =

[ 1/2*a+1/2*d+1/2*(a^2-2*a*d+d^2+4*b*c)^(1/2), 0]

[ 0, 1/2*a+1/2*d-1/2*(a^2-2*a*d+d^2+4*b*c)^(1/2)]

 

Snum = [-1,1;1,0]'

Snum =

-1 1

1 0

 

Dnum = [2, 0; 0, 3]

Dnum =

2 0

0 3

 

Anum = Snum * Dnum * inv(Snum)

Anum =

3 1

0 2

 

[Vnum,Dnum] = eig(Anum)

Vnum =

1 -0.70711

0 0.70711

Dnum =

3 0

0 2