The Geometry of Diagonalization

We start with a diagonal matrix:

In[1]:=

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

Here is a Quicktime animation that illustrates how a diagonal matrix transforms a square.

Now we form a matrix B that is similar to A using the matrix P.

In[2]:=

P = (2 1) ; 1 1

In[3]:=

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

Out[3]=

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

Note that B has the same eigenvalues as A.

In[4]:=

Eigenvalues[B]

Out[4]=

{9/10, 10/9}

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

In[5]:=

Det[B] == Det[A]

Out[5]=

True

In[6]:=

Det[B]

Out[6]=

1

However, after running this Quicktime animation, it looks as if B transforms the plane in a more complicated fashion than A does.

But if we use the right choice of coordinates, this Quicktime animation illustrates that B transforms the plane in the "same" way as A.

Converted by Paul Blanchard using Mathematica  (November 24, 2002)