Jordan Form

Question 1

Jordan Form

Answer 1

Diagonal except the possibility of a 1 above the diagonal in between two eigenvalues which are equal.

Question 2

The number of the exponent in m(x) tells you..

Answer 2

the size of the biggest jordan block

Question 3

jordan block

Answer 3

a block with all the same eigenvalues on the diagonal and 1s above the diagonal. Jordan form is made of multiple (or just 1) of these.

Question 4

How to find the jordan form of a matrix

Answer 4

1. find the characteristic polynomial
2. find the minimum polynomial
3. work out the eigenspaces of the eigenvalues
4. work out the generalised eigenspaces
5. extend the basis of the generalised eigenspace to a basis of the whole space
6. work out this as a basis a,b,c,… of A
7. calculate Aa, Ab, Ac, … and then put these values into a matrix (Aa Ab Ac …)