Unit 3

Question 1

For standard VQ function, under what circumstances will x* yield a unique global minimum?

Answer 1

If A is positive definite

Question 2

What determines if a square nxn matrix A is positive semi-definite?

Answer 2

A square nxn matrix A is said to be positive semi-definite if, for all nx1 vectors x, the scalar value x^TAx>=0

Question 3

See

Answer 3

First 3 lines of notes on possible definite matrices

Question 4

3 points regarding PSD and PD matrices?

Answer 4

1) set of PD matrices is a proper subset of PSD matrices
2) being PD is a sufficient condition for being PSD
3) being PSD is a necessary condition for being PD

Question 5

When is an nxn matrix A said to be positive definite?

Answer 5

A square nxn matrix A is said to be positive definite if (for all x is a subset of R^n) such that x is not equal to 0n, it is true that x^TAx>0

Question 6

Note

Answer 6

Since no matrix can satisfy x^TAx>0 if x=0n we exclude it from our definition

Question 7

What is a 1x1 matrix positive definite?

Answer 7

If it’s element is positive

Question 8

Quick rule for checking if a diagonal matrix is positive definite? And why?

Answer 8

If all of it’s entries are positive, it is positive definite because a positively weighed sum of squares can never be negative

Question 9

How to tell if a matrix A is negative definite, and what this means for f(x)?

Answer 9

If -A is positive definite, then A is negative definite - tf if A has this property f(x) will have a unique global maximum (if f(x)=x^TAx only then UGM at origin)

Question 10

How to tell if matrix A is negative semi-definite?

Answer 10

If -A is positive semi-definite, then A is negative semi-definite

Question 11

What is an indefinite matrix?

Answer 11

One that isn’t PSD, PD, NSD or ND

Question 12

Why would a matrix come out as indefinite?

Answer 12

A matrix comes out as indefinite when x^TAx can be either positive or negative depending on the vector x chosen

Question 13

Different types of matrix A and what this means regarding maximum and minimum points? (5)

Answer 13

PD - unique global min at x*
PSD - global min at x*
Indefinite - no min/max
ND - unique global max at x*
NSD - global max at x*

Question 14

Necessary condition for PD matrix?

Answer 14

A positive determinant

Question 15

Note:

Answer 15

Is PD all diagonal entries will be positive

If PSD all diagonal entries will be non-negative

Question 16

3 points regarding PSD, PD and indefinite matrices wrt eigenvalues?

Answer 16

If PD AND symmetric, all eigenvalues are positive

If PSD AND symmetric, all EVs are nonnegative

If indefinite AND symmetric, both positive and negative EVs

Question 17

Key point regarding the determinant test for positive definiteness?

Answer 17

If used in exam, MUST point out can only be done due to matrix being symmetric

Question 18

Explain the determinant test for positive definiteness?

Answer 18

See notes

Question 19

How can the determinant test be altered for testing if a matrix is PSD?

Answer 19

If ALL principal minors are non-negative then the matrix is PSD (NOT just the leading principal minors) (see example 9)