Math 2406: Abstract Vector Spaces

Lecture 22 Plan  (Thursday, November 8, 2007).

Apostol 6.12-6.13.

We've been discussing similar matrices.  For a given matrix A, one very important similar matrix B = C^{-1}AC is the diagonal matrix with the eigenvalues of A on the diagonal. What is the transition matrix C in this case? It is the matrix with columns equal to eigenvectors of A.
Then we'll discuss the Cayley-Hamilton Theorem, which states that any matrix satisfies its own characteristic equation.

Review questions:

1. Why is it important to have complex scalars for the Cayley-Hamilton Theorem?
2. Why is  a linear transformation diagonal when one represents it in the basis of its eigenvectors?