Chapter 4 Topics in Linear Algebra Part 2

Question 1

What is the jordan block?

Answer 1

Jordan block of order r associated with eigenvalue Lamda is and rxr matrix : With lamda along the diagonal and 1’s on the super diagonal and zero’s elsewhere

Question 2

What does jordan block of order 1 with eigenvalue 7 look like?

Answer 2

(7)

Question 3

What does jordan block of order 2 with eigenvalue 7 look like?

Answer 3

(7 1

0 7)

Question 4

What type of matrix is a jordan block known as?

Answer 4

Upper triangle matrix - special case

Question 5

What happens when Jr(0) is raised to power

Answer 5

Everytime its multipled by itself it moves the 1’s across one column each time.

Question 6

How many eigenvectors does a Jordan Block have?

Answer 6

1 linearly independent eigenvector as there is only one eigenvalue

Question 7

What are the properties of a direct sum matrix P+Q

Answer 7

P is m x m and Q is n x n so P+Q is M+n x m+n
The characteristic polynomial of the direct sum in the product of the characteristic polynomials of P and Q
Eigenvalues of direct sum is joining the lsit of eigenvalues of P and Q

Question 8

What is a theorem connecting similar and jordan blocks

Answer 8

Every real n x n matrix is similar to a direct sum of jordan blocks. This sum is unique up to the order of the Jordan blocks and is denoted J(A) called the jordan canonical form of A

Question 9

What si the significant of the jordan canonical form of A

Answer 9

We are no longer limited to diagonalisable matrices to learn about A ^k easily as we can have A ^k = T X ^kT^-1 where X is J(A)

Question 10

Define a partition

Answer 10

A partitions of positive integer K is an expression of K as a sum of positive integers. Usually we write: K = c1+…+cn where n>=1 and c1>=….>=cn>=1

Question 11

Explain a monic polynomial

Answer 11

An=1 the coefficient of the biggest power is 1

Question 12

What is the minimal polynomial of A (nxn)

Answer 12

Its the monic polynomial such that P(A)=0 with the lowest degree posssible

Question 13

What does similar matrices tell you about their minimal polynomial

Answer 13

They are the same

Question 14

Explain the cayley hamilton theorem

Answer 14

If A is n x n let p(x) = characteristic polynomial and we will obtain P(A) =0
So for every square matrix there is a polynomial where P(a) = 0 but is it the minimal?

Question 15

What is the relationship between the characteristic equation and the minimal polynomial

Answer 15

Characteristic polynomial is not always the minimal polynomial

Question 16

How many possible jordan canonical forms are there for a matrix A

Answer 16

Each distinct eigenvector has a certain number of possible Direct sums of Jordan blocks - the amount for each distinct eigenvalues multipled together gives total options

Question 17

Out of all possible jordan canonical forms of A how do I know which one is correct

Answer 17

Minimal polynomial of A and J(A) Are the same as they are similar