Unit 3 Flashcards
For standard VQ function, under what circumstances will x* yield a unique global minimum?
If A is positive definite
What determines if a square nxn matrix A is positive semi-definite?
A square nxn matrix A is said to be positive semi-definite if, for all nx1 vectors x, the scalar value x^TAx>=0
See
First 3 lines of notes on possible definite matrices
3 points regarding PSD and PD matrices?
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
When is an nxn matrix A said to be positive definite?
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
Note
Since no matrix can satisfy x^TAx>0 if x=0n we exclude it from our definition
What is a 1x1 matrix positive definite?
If it’s element is positive
Quick rule for checking if a diagonal matrix is positive definite? And why?
If all of it’s entries are positive, it is positive definite because a positively weighed sum of squares can never be negative
How to tell if a matrix A is negative definite, and what this means for f(x)?
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)
How to tell if matrix A is negative semi-definite?
If -A is positive semi-definite, then A is negative semi-definite
What is an indefinite matrix?
One that isn’t PSD, PD, NSD or ND
Why would a matrix come out as indefinite?
A matrix comes out as indefinite when x^TAx can be either positive or negative depending on the vector x chosen
Different types of matrix A and what this means regarding maximum and minimum points? (5)
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*
Necessary condition for PD matrix?
A positive determinant
Note:
Is PD all diagonal entries will be positive
If PSD all diagonal entries will be non-negative
3 points regarding PSD, PD and indefinite matrices wrt eigenvalues?
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
Key point regarding the determinant test for positive definiteness?
If used in exam, MUST point out can only be done due to matrix being symmetric
Explain the determinant test for positive definiteness?
See notes
How can the determinant test be altered for testing if a matrix is PSD?
If ALL principal minors are non-negative then the matrix is PSD (NOT just the leading principal minors) (see example 9)
