Jordan Normal Form Flashcards
Jordan Normal Form
Description
-let A be an nxn matrix in C
-there exists an invertible matrix P such that
P^(-1)AP = J
-where J is in Jordan normal form
-a matrix in Jordan normal form is made up of a series of Jordan blocks, with all other entries in the matrix being zero
Jordan Block
- an nxn matrix with entries λ along the main diagonal
- a 1 above each entry on the main diagonal
- and zeros everywhere else
- this can mean a 1x1 matrix with a single entry λ
Generalised Eigenvector Definition
-let A be a square matrix
-let λ be an eigenvalue of A and let v be a non-zero eigenvector of A associate with λ, i.e. Av = λv
-a vector w is called a generalised eigenvector if:
Aw = λw + v
Generalised Eigenvector
2x2 Jordan Block
-if the Jordan block A has λ and 1 in the first row and 0 and λ in the second row, then:
A ^e1 = λ ^e1 , so ^e1 is an eigenvector of A,
and A ^e2 = λ ^e2 + ^e1 so ^e2 is a generalised eigenvector
When to look for generalised eigenvectors?
-when working over the field of complex number, C, look for generalised eigenvectors if the geometric multiplicity is smaller than the algebraic multiplicity
How to put a matrix into Jordan normal form?
1) find the characteristic polynomial
2) find the eigenvalues and corresponding algebraic multiplicities
3) find the associated eigenvectors and geometric multiplicities
4) if, for any of the eigenvalues, the geometric multiplicity is smaller than the algebraic multiplicity then for that eigenvalue you need to find a generalised eigenvector
5) Use the formula
Av = λi v + vi to find the generalised eigenvector v corresponding to λi
6) For an nxn matrix you should now have n linearly independent eigenvectors and generalised eigenvectors, the matrix J in Jordan normal form has ith column with the coordinates of A*vi using the n vectors as a basis
7) The matrix P such that P(-1)AP = J has the n vectors as its columns
