Chapter 9 - Quadratic Forms, Orthogonal And Symmetric Matrices

Question 1

Properties of dot product ( inner product)

Answer 1

Symmetry : x • y = y • x
Bilinearity: for all x,y,x’,y’ in R^n and s,t (real numbers),
(sx + tx’) • y = sx • y + x’ • y
x • (sy + ty’) = sx • y + x • y’
Non-negativity: x • x > or equal to 0
|x| = sqrt (x • x). This is the length of x
x = 0 iff |x| = 0
If x doesn’t equal 0, then x/|x| has length 1

Question 2

Orthogonal vectors x, y

Answer 2

If x • y = 0

Question 3

Orthonormal set of vectors

Answer 3

If vi • vi = delta (ij) = 1 if i=j
0 if i doesn’t equal j
Delta (ij) is the Kornecker delta function

Question 4

Orthogonal matrix

Answer 4

Ax • Ay = x • y for all x, y in R^n
Has an inverse A^-1 = A^T which is also orthogonal
A^T * A = E
If A and B are orthogonal, so is AB
Det(A) = 1 or -1

Question 5

Orthogonal group

Answer 5

The set of all orthogonal n by n matrices
O (n) = {A in Mn (R) | A^T * A = E
A^-1 = A^T

Question 6

A is in O (n) if

Answer 6

Its columns (considered as a set of vectors) form an orthonormal set

Question 7

Symmetric matrix

Answer 7

If Ax • y = x • Ay for all x, y in R^n
All eigenvalues of A are real numbers
If Au = lamdau and Av = muv and lamda doesn’t equal mu, then u • v = 0
(i.e. if u and v are eigenvectors of A with distinct eigenvalues, then they are orthogonal)

There is an orthogonal matrix P and a diagonal matrix D such that A = PDP^-1 = PDP^T

Question 8

Diagonal matrix

Answer 8

A square matrix with zero entries in every position that isn’t on the leading diagonal

Question 9

Quadratic form as a matrix equation

Answer 9

Q (x) = x^T * B * x

Question 10

Rank of Q

Answer 10

r + s

Question 11

Signature of Q

Answer 11

r - s

Question 12

Positive definite

Answer 12

If Q (x) is positive for all values of x not equal to 0

If the eigenvalues (of B) are positive

Question 13

Negative definite

Answer 13

If Q (x) is negative for all values of x not equal to 0

If the eigenvalues (of B) are negative

Question 14

Positive semidefinite

Answer 14

If Q (x) > or = to 0 for all values of x not equal to 0

If the eigenvalues (of B) are > or = to 0

Question 15

Negative semidefinite

Answer 15

If Q (x) < or = to 0 for all values of x not equal to 0

If the eigenvalues (of B) are < or = to 0

Question 16

Indefinite

Answer 16

If Q (x) can take both positive and negative values

If there is at least one positive and at least one negative eigenvalue (of B)

Question 17

Taylor series

Answer 17

f (x0 + deltax) = f (x0) + (1/2) (deltax)^T * H (deltax)
= f (x0) + (1/2)Q (deltax)

where Q (delta*x) is a quadratic form associated with the hessian matrix H

Question 18

Hessian matrix

Answer 18

H = matrix of:
[Fxx fxy]
[Fxy fyy]

Question 19

Stattionary point is local minimum

Answer 19

If the eigenvalues of H are positive

Question 20

Local maximum stationary point

Answer 20

If the eigenvalues of H are negative

Question 21

Saddly point stationary point

Answer 21

If at least one eigenvalue of H is positive and at least one is negative

Question 22

Method to find eigenvalues and eigenvectors

Answer 22

Work out A-lamda*Identity matrix
Find determinant of this
Put that equal to 0
Factorise to find values of lamda
Sub these values back into the 1st step
Normalise eigenvectors

Question 23

To normalise an eigenvector (a,b)

Answer 23

Do 1/(sqrt of (a^2 + b^2) (a,b)