**Matrix diagonalisation** To diagonalise a matrix A one needs to find another matrix C to bring A in its diagonal form D. One also needs the inverse of matrix C, C

^{-1}:

D=C

^{-1}AC

To find matrix C the eigenvalues and eigenvectors of A need to be computed first. This can be done using the characteristic polynomial F:

F(x)=det(A-xI)=0 where I is the unit matrix and det is short for determinant.

We continue by example. Let

` `

A = 2 6

0 -1

F(x) = det( 2-x 6

0 -1-x)

= (2-x)(-1-x)-6.0 = (2-x)(-1-x) =0

Hence the eigenvalues are x=2 and x=-1.

To find the corresponding eigenvectors, substitute the eigenvalues back into A-xI=0.

The two eigenvectors are (1,0) and (-2,1). Arranging these into a matrix gives matrix C:

` `

C = 1 -2

0 1

Its inverse is

C^{-1} = 1 2

0 1

Now we can calculate the diagonal matrix D from: D=C

^{-1}AC

` `

D = (1 2) (2 6) (1 -2)

(0 1) (0 -1) (0 1)

D = (1 2) (2 2)

(0 1) (0 -1)

D = (2 0)

(0 -1)

__See also:__ https://www.math.okstate.edu/~binegar/3013-S99/3013-l16.pdf