# Homework Help: A certian Linear Algebra gimmick needed for a part of my project

1. Jan 14, 2014

### aashish.v

1. I need to prove that for any matrix A(n,n) and a vector v(n,1) the following is true...

vTAv=vTATv

So far I wasn't able to think of anyway for proving this... any help will be appreciated.

2. Jan 14, 2014

### aashish.v

I do understand intuitively how this can be true...
Because of the arrangement of the equation, both side of the equation gets the same value.
I am able to show that with small example,,

$$A=\left[\begin{array}{cc} p & q\\ r & s \end{array}\right],v=\left[\begin{array}{cc} a & b\end{array}\right] \Rightarrow v'Av=v'A'v=a^{2}p+abr+abq+b^{2}s$$

But I need rigorous proof for the same...

Thank you. :)

3. Jan 14, 2014

### ehild

What is the dimension of vTAv and vTATv?
What is (vTAv)T? How do you transpose a matrix product?

ehild

4. Jan 14, 2014

### Dick

For two matrices A and B, $\left(AB\right)^T=B^TA^T$. If you don't already know that prove it using sigma notation.

5. Jan 14, 2014

### aashish.v

So this is what I understand...

since the equation $$v'Av$$ is a number... we can write...

$$v'Av=(v'Av)'=v'(v'A)'=v'A'v$$

Am I correct?

6. Jan 14, 2014

### ehild

Yes, that was I wanted you to do.

ehild

7. Jan 14, 2014

### Staff: Mentor

aashish.v, when you post a problem, include your work in the inital post. Without the work you show in your 2nd post, your 1st post would normally earn you a warning for an unacceptable homework request.