# Solve this eigen value problem

1. Oct 2, 2008

### sayan2009

plz solve this eigen value problem

A nd B are matrices of order n*n.now it is given that sum of each row of A is 2 nd that of B is 1...then show that 2 is an eige value of the product matrix AB

2. Oct 4, 2008

Re: plz solve this eigen value problem

Let $$\vec v$$ be an $$n \times 1$$ vector like this:

$$\vec v' = \left[\frac 1 n \frac 1 n \dots \frac 1 n \right]$$

Then compute all of

\begin{align*} & A \vec v \\ & B \vec v\\ & (AB) \vec v \end{align*}

and remember that for any matrix $$W$$ and vector $$\vec z$$, if there is a scalar $$k$$ such that

$$W \vec z = k \vec z$$

then $$k$$ is an eigenvalue of the matrix $$W$$.

Last edited: Oct 4, 2008
3. Oct 7, 2008

### sayan2009

Re: plz solve this eigen value problem

how to compute (AB)v

4. Oct 7, 2008

Re: plz solve this eigen value problem

Here is a small example (note: the rows in this matrix do not sum to either 1 or 2, as doing that would be solving the problem for you. However, precisely the same steps work)

\begin{align*} A &= \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix}\\ \vec v & = \begin{bmatrix} \frac 1 2 & \frac 1 2 \end{bmatrix}' \end{align*}

Then
$$A \vec v = \begin{bmatrix} 1 & 2 \\ 3 & 4 \end{bmatrix} \begin{bmatrix} 1/2 \\ 1/2 \end{bmatrix} = \begin{bmatrix} {(1+2)}/2 \\ {(3+4)}/2 \end{bmatrix}$$

so in this case, and in every case, the product $$A \vec v$$ has as its entries the means of the rows of $$A$$.

5. Oct 7, 2008

### sayan2009

Re: plz solve this eigen value problem

but why should i try to get mean here????i mean i can take v (transpose) as[1 1 1 .... 1](n times)
why r u taking [1/n 1/n ......... 1/n]?????

6. Oct 7, 2008

### Dick

Re: plz solve this eigen value problem

Go ahead. Just use [1,1,1...]. (Not that there's anything wrong with using [1/n,1/n,...], you'll get the same result in the end).

7. Oct 7, 2008

Re: plz solve this eigen value problem

Just try the multiplication (or use Dick's suggestion) and notice how the result compares to the vector $$\vec v$$.

Remember that if $$A \vec v = k \vec b$$ then $$k$$ is an eigenvalue of the matrix $$A$$.

8. Oct 7, 2008

### sayan2009

Re: plz solve this eigen value problem

so the solution is using v=[1 1 1 1 ... 1]
we can easily get Av=2v & Bv=1v
then(AB)v=A(Bv)=A(1v)=Av=2v
so 2 is an eigen value of AB................
is this right solution?????????????????????

9. Oct 7, 2008

### Dick

Re: plz solve this eigen value problem

(AB)v=2v. That sure looks like it says 2 is an eigenvalue to me.

10. Oct 7, 2008

### sayan2009

Re: plz solve this eigen value problem

is that solution correct man???????????

11. Oct 7, 2008

### Dick

Re: plz solve this eigen value problem

Do you have any doubts??????? You don't need me to approve your solution. If you believe in it go for it.

12. Oct 10, 2008

### HallsofIvy

Staff Emeritus
Re: plz solve this eigen value problem

Surely you meant $$A\vec v= k \vec v$$?
Which is the definition of "eigenvalue".