## Optimization 2 - Homework 2

## Homework Problems

### #2

Let be an arbitrary matrix, and let be itโs symmetric counterpart. Show for all if and only if is positive definite.

- WTS is positive definite
- Suppose is positive definite
- Then for all (defn of positive definite)
- Then (rewrite )
- Because , eigenvalues of are positive (property of positive definite)
- Then the eigenvalues of either or are positive
- Eigenvalues of are the eigenvalues of (property of eigenvalues)
- Then the eigenvalues of and must both be positive
- Because the eigenvalues of are positive, (property of positive definite)
- Then (definition of positive definite)
- Done

- suppose
- Then is positive definite
- Then the eigenvalues of (and simmilarly ) are positive (property of positive definite)
- Then the eigenvalues of and are positive (property of eigenvalues)
- Then the eigenvalues of are positive (property of eigenvalues)
- Then the eigenvalues of are positive (rewrite )
- Then is positive definite (property of positive definite)
- Done

- Then is positive definite
- Done

### #3

Let be a symmetric matrix. Assume that there exist a matrix such that .

Show that is positive semidefinite.

- The eigenvalues of are non-negative, as for each eigenvalue
- Non-negative eigenvalues implies (property of positive semidefinite)
- Done

Show that if has full column-rank, then is positive definite.

- Suppose has full column-rank
- Then the columns of are linearly independent (defn of full column-rank)
- Then is nonsingular (property of symmetric matrices)
- Then has nonzero eigenvalues (property of nonsingular matrices)
- Because is positive semi-definite, it has non-negative eigenvalues.
- Then the eigenvalues of must be strictly positive.
- Then is positive definite (property of positive definite)
- Done

### #4

Let be a symmetric matrix. If are the (real) eigenvalues of , show that the Rayleigh Quotient satisfies for and

- Eigenvectors corresponding to distinct eigenvalues of A are orthogonal to each other (property of symmetric matrices)
- Write
- Write
- Write
- Write
- Then

Show that the quadratic form is

coerciveif and only if A is positive definite.

- A quadratic form is coercive if there exists a constant such that for all
- WTS is positive definite is coercive
- Suppose is coercive
- Then there exists such that for all
- Then for all (divide both sides by )
- From before, we know is greater than or equal to the smallest eigenvalue of .
- Then the smallest eigenvalue of must be positive
- Then all eigenvalues of must be positive
- Then is positive definite

- Suppose is positive definite
- Then the eigenvalues of are positive
- From before, we know is greater than or equal to the smallest eigenvalue of .
- let represent the smallest eigenvalue of
- Then for all (multiply both sides by )
- Because is positive definite, is positive
- Then is coercive with positive

Show that if and , then the quadratic function is coercive if and only if is positive definite.

- A scalar function is coercive if

### #5

To approximate a function over an interval by a polynomial of degree n we minimize the integral where . Find the equations satisfied by the optimal coefficients to minimize . Write your answer in terms of matrix , vector , and scalar given by

, ,

- WTS a system of equations to solve for involving , and
- The optimal is such that
- Then the optimal derivatives of the value of with respect to is
- Note that

- Then we have
- Note that
- Then
- Then Ac = b
- done