Diagonalisation

Question 1

When is a square matrix B similar to a square matrix A?

Answer 1

Iff there exists an invertible matrix such that B = P^-1AP

Question 2

What is the geometric interpretation of B?

Answer 2

B is the same linear transformation as A, but represented in a basis defined by the column vectors of P.

Question 3

How do we represent a vector transformed by the mapping?

Answer 3

APz (where z is the vector).

Question 4

How would we expressed the vector in the new basis?

Answer 4

We must multiply by P^-1, obtaining P^-1AP*z.

Question 5

When is a square matrix A (n*n) diagonalisable?

Answer 5

Iff there is exists an invertible matrix P such that D = P-1A*P.

Iff its eigenvectors form a basis of n dimensional space (i.e. n independent eigenvectors).

Iff the geometric order of each eigenvalue = its algebraic order.