The Geometry of Diagonalization

We start with a diagonal matrix:

D = (9 ) ; -- 10 0 10 -- 0 9

Here is a Quicktime animation (alternate version) that illustrates how a diagonal matrix transforms a square.

We form a matrix A that is similar to D using the matrix P.

P = (2 1) ; 1 1

A = P . D . Inverse[P] ; MatrixForm[A]

( 31 19 ) -- -- 45 45 19 119 --- --- 90 90

Note that A has the same eigenvalues as D.

Eigenvalues[A]

{9/10, 10/9}

Therefore, A transforms area by the same factor as D.

Det[A] == Det[D]

True

Det[A]

1

However, it looks as if A transforms the plane in a more complicated fashion than D does.

But if we use the right choice of coordinates, A transforms the plane (alternate version) in the "same" way as D.

Converted by Paul Blanchard using Mathematica  (November 11, 2008)