Week 8 - Orthogonal diagonalisation, Quadratic forms

Question 1

An orthogonal set of non-zero vectors is…

Answer 1

linearly independent!

Question 2

An nxn matrix P is orthogonal if…

Answer 2

P^T P = In (identity matrix)

The columns of P form an orthogonal set of UNIT VECTORS

Question 3

Theorem / 3 checks for orthogonal diagonalisation

Answer 3

1. P^T P = In
2. P^-1 = P^T (invertible and = identity matrix)
3. P^T AP = D (diagonal matrix)

Question 4

Condition for a square matrix to be orthogonally diagonalisable

Answer 4

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

Question 5

How to solve/sketch the contour for a quadratic form x^T Ax =2?

Answer 5

1. Let x = Pz where z = (X Y)
2. z^T Dz = {expand & write out full quadratic form}
3. Determine if ELLIPSE or HYPERBOLA
4. 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)
5. Find the X and Y intercepts
6. Connect the points

Question 6

How to test for semidefiniteness?

Answer 6

• If A has an eigenvalue = 0, |A|=0 so principal minors test fails.
• Must find the eigenvalues of A and get >=1 zero evalue

Question 7

How to show that B^T B is symmetric?

[2022 IRDAP paper]

Answer 7

B^T B is symmetric as (B^T B)^T = B^T B