# Homework Help: What are the matrices B, C, D?

1. Aug 11, 2011

### squenshl

1. The problem statement, all variables and given/known data
Let x = (x1,x2) $\in$ Rn, x1 $\in$ Rn1, x2 $\in$ Rn2, n1 + n2 = n and A $\in$ Rnxn be symmetric and positive definite.
a) Let x0 $\in$ Rn. Show that we can write (x-x0)TA(x-x0) = ||L(x-x0||22. Is L unique?
b) Consider the quadratic term b = xTAx. Show that we can write b = x1TBx1 + 2x1TCx2 + x2T
Dx2, and what are the matrices B, C, D?
2. Relevant equations

3. The attempt at a solution
a) Is it using the definition of symmetric & positive defintie matrices.
b) Isn;t that just a quadratic form?

2. Aug 11, 2011

### stringy

For (a), perhaps you should look into the square root of a matrix and under what conditions it is unique.

For (b), you can solve this by partitioning A into appropriately sized blocks and carrying out block multiplication. And remember that A is symmetric! You'll need that fact to finish the last step.

3. Aug 11, 2011

### Ray Vickson

Hint for (a): Look at *Cholesky Factorization* (Google search).

RGV

4. Aug 14, 2011

### squenshl

Never done Cholesky factorization before in my life.

5. Aug 14, 2011

### squenshl

Do you mean the square root of positive definite symmetric matrix?
Not sure what you mean for b)?

6. Aug 14, 2011

### Ray Vickson

So what? I gave a suggestion and it is up to you to take the advice or not.

RGV

7. Aug 15, 2011

### squenshl

A little stuck on b), what do you partitioning A into appropriately sized blocks and carry out block multiplication.

8. Aug 15, 2011

### stringy

You can split matrices into blocks (they must be the appropriate sizes so that the multiplication is defined) and multiply them. It's quite helpful in some proofs and helps with notational issues.

Look http://en.wikipedia.org/wiki/Block_matrix#Block_matrix_multiplication".

And yes, look into the square root of a positive semidefinite matrix. BTW, it's related to the factorization that Ray Vickson mentioned. Depending on where you read up on this, you might see that factorization and square roots mentioned in the same chapter/article/section, etc.

Last edited by a moderator: Apr 26, 2017
9. Aug 17, 2011

### squenshl

b = xTAx = (x1 x2)TA(x1 x2)
where A has entries B = a11 C = a12 = a21 (symmetric matrix) and D = a22
b = x1TBx1 + x1TCx2 + x2TCx1 + x2TDx2
= x1TBx1 + 2x1TCx2 + x2TDx2

But how do we define B, C, D?
Is it just similar to the wiki page?

Last edited: Aug 17, 2011
10. Aug 17, 2011