What are the matrices B, C, D?

[squenshl]

Homework Statement

Let x = (x₁,x₂) $\in$ Rⁿ, x₁ $\in$ R^n₁, x₂ $\in$ R^n₂, n₁ + n₂ = n and A $\in$ R^nxn 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 = x^TAx. Show that we can write b = x₁^TBx₁ + 2x₁^TCx₂ + x₂^T
Dx₂, and what are the matrices B, C, D?

Homework Equations

The Attempt at a Solution

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

 

[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.

 

[Ray Vickson]

Homework Statement

Let x = (x₁,x₂) $\in$ Rⁿ, x₁ $\in$ R^n₁, x₂ $\in$ R^n₂, n₁ + n₂ = n and A $\in$ R^nxn 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 = x^TAx. Show that we can write b = x₁^TBx₁ + 2x₁^TCx₂ + x₂^T
Dx₂, and what are the matrices B, C, D?

Homework Equations

The Attempt at a Solution

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

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

RGV

 

[squenshl]

Never done Cholesky factorization before in my life.

 

[squenshl]

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

 

[Ray Vickson]

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

RGV

 

[squenshl]

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

 

[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.

 

[squenshl]

b = x^TAx = (x₁ x₂)^TA(x₁ x₂)
where A has entries B = a₁₁ C = a₁₂ = a₂₁ (symmetric matrix) and D = a₂₂
b = x₁^TBx₁ + x₁^TCx₂ + x₂^TCx₁ + x_2^TDx₂
= x₁^TBx₁ + 2x₁^TCx₂ + x_2^TDx₂

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

 

[squenshl]

Any ideas?