Week 8 - Orthogonal diagonalisation, Quadratic forms Flashcards
1
Q
An orthogonal set of non-zero vectors is…
A
linearly independent!
2
Q
An nxn matrix P is orthogonal if…
A
P^T P = In (identity matrix)
The columns of P form an orthogonal set of UNIT VECTORS
3
Q
Theorem / 3 checks for orthogonal diagonalisation
A
- P^T P = In
- P^-1 = P^T (invertible and = identity matrix)
- P^T AP = D (diagonal matrix)
4
Q
Condition for a square matrix to be orthogonally diagonalisable
A
If and only if it is SYMMETRIC, ie. A^T = A
Need an orthogonal matrix P such that P^T AP = D where D is a diagonal matrix
5
Q
How to solve/sketch the contour for a quadratic form x^T Ax =2?
A
- Let x = Pz where z = (X Y)
- z^T Dz = {expand & write out full quadratic form}
- Determine if ELLIPSE or HYPERBOLA
- Find the unit vector in the big X and big Y directions & sketch the X and Y axes (on top of the x and y axes)
- Find the X and Y intercepts
- Connect the points
6
Q
How to test for semidefiniteness?
A
- If A has an eigenvalue = 0, |A|=0 so principal minors test fails.
- Must find the eigenvalues of A and get >=1 zero evalue
7
Q
How to show that B^T B is symmetric?
[2022 IRDAP paper]
A
B^T B is symmetric as (B^T B)^T = B^T B