Chapter 7 Rank 1 perturbation Brauers

Question 1

Give two ways to use real numbers to manipulate matrix eigenvalues

Answer 1

If I am manipulating the eigenvalues of A I can use:
A + bI
or
bA where b is a real number

Question 2

Why would we want to manipulate eigenvalues

Answer 2

As we want the ideal situation to find the limit of a matrix where we are dealing with a positive stochastic matrix ideally with fast convergence

Question 3

How do the eigenvalue of AB and BA relate?

Answer 3

They have the same non zero eigenvalues

Question 4

What do we know about a rank one stochatic matrix?

Answer 4

K has all entries 1/m so eigenvalues are 1 with multiplicity 1 and 0 with multiplicty m-1

Question 5

Any rank one matrix of the for 1/n (e)(e^transpose) or ew^tranpose where (w^tranpose)(e)=1 has limit of what?

Answer 5

ew^tranpose … aka itself

Question 6

State brauers theorem

Answer 6

Let A be any nxn matrix with eigenvalues k,,…,kn and v being the eigenvector Av =kv
Let B be the nxn rank 1 matrix vy^tranpose.
Then the eigenvalues of A + B are the same as the
eigenvalues of A except k1 is replace by k1+ y^tranpose(v). Eigenvalues: k, + y^tranpose(v) ,…, kn (only 1 transformed)

Question 7

If i multiply matrix A by b what are the eigenvalues of bA

Answer 7

B multiplied by eigenvalues of A