02 Random Variables: Matrix Algebra

Question 1

If matrix Aₙₓₚwhat are i and j in the matrix elements?

Answer 1

aᵢⱼ

• > i = 1….n
• > j = 1…p

Question 2

What elements does the transpose of Matrix A have?

Answer 2

aⱼᵢ

Question 3

What is a square matrix?

Answer 3

n = p

rows = # columns

Question 4

What is a diagonal matrix?

Answer 4

diag(aᵢᵢ) has numbers on the main diagonal (aᵢᵢ) and 0’s on the off-diagonal elements

Question 5

What is the identity matrix Iₚ?

Answer 5

a diagonal matrix with p elements on the main diagonal, which are all 1

Question 6

What is a unit matrix?

Answer 6

A unit matrix is an integer matrix consisting of all 1s

Question 7

What is a symmetric matrix?

Answer 7

It is symmetric around the diagonals such that Aᵀ = A

Question 8

What is an idempotent matrix?

Answer 8

a matrix which, when multiplied by itself, yields itself. must be a square matrix. e.g. I₃
A = A ⋅ A

Question 9

What is an orthogonal matrix?

Answer 9

AᵀA = AAᵀ = Iₚ

Question 10

What is an inverse matrix?

Answer 10

A⁻¹A = A A⁻¹ = Iₚ

Question 11

How to multiply two matrices?

Answer 11

1) check the dimensions. Inner numbers must be the same, outer numbers give dimensions of new matrix
2) multiply each first row element of the first matrix with the corresponding element of the first column of the second matrix. Sum up.
3) Repeat for all columns and rows.

Question 12

What is the determinant |A| of a 2x2 matrix?

Answer 12

the multiplied downward diagonal - multiplied upward diagonal, i.e.:
a₁₁⋅a₂₂ - a₂₁⋅a₁₂

Question 13

What is the determinant |A| of a 3x3 matrix?

Answer 13

each element of the first row multiplied with the determinant of the second & third row of the remaining columns, i.e. a₁₁⋅ | a₂₂ a ₂₃ /// a₃₂ a₃₃| - a₁₂ |…| + a₃₂ |…|

Question 14

What is the trace of a matrix?

Answer 14

sum of all fiagonal elements

Question 15

What is the matrix property A ( B + C) …?

Answer 15

A ( B + C ) = AB + AC

Question 16

What is the matrix property (AB)ᵀ …?

Answer 16

(AB)ᵀ = Bᵀ Aᵀ

Question 17

What is the matrix property (ABC)ᵀ…?

Answer 17

(ABC)ᵀ = Cᵀ Bᵀ Aᵀ

Question 18

What is the spectral decomposition of A?

Answer 18

ΓΛΓᵀ

Question 19

What is Λ?

Answer 19

the matrix of eigenvalues = diag (λ₁, λ₂ , … λₚ)

where λ₁ ≥ λ₂ ≥ … ≥ λₚ

Question 20

What is Γ?

Answer 20

the matrix of eienvectors

Question 21

What does tr(A) have to do with Λ?

Answer 21

tr(A) = tr(Λ) = Σ λᵢ

Question 22

What does | A | have to do with Λ?

Answer 22

A | = | Λ | = π λᵢ

Question 23

What does Aᵃ have to do with spectral decomposition?

Answer 23

Aᵃ = ΓΛᵃΓᵀ

Question 24

What is Q(x)?

Answer 24

the quadratic form of A = xᵀ A x, where:
xᵀ = 1 x p
A = p x p
x = p x 1

Question 25

When is A positive definite and what are the implications?

Answer 25

1) Q(x) > 0 for every x
2) A⁻¹ exists
3) λᵢ > 0

Question 26

When is A positive semi-definite?

Answer 26

When Q(x) ≥ 0 for every x

Question 27

What does negative definite mean?

Answer 27

Q(x) < 0 for every x

Question 28

What is the first derivative of the quadratic form and what does it mean?

Answer 28

1) 2Ax
2) 1st derivative gradient
3) vector of first derivatives

Question 29

What is the second derivative of the quadratic form and what does it mean?

Answer 29

1) 2A

2) Hessian Matrix

Question 30

What is the distance between two vectors x and y?

Answer 30

d(x, y)ₐ = √ [ (x-y)ᵀ A (x-y) ]

Question 31

What is the norm of a vector x?

Answer 31

the distance from x to zero:

||x||ₐ = d(x, 0)ₐ = √ [ xᵀ A x ]

Question 32

What is the angle between two vectors x and y?

Answer 32

(Cosθ)ₐ = [ xᵀ A y ] / ||x||ₐ ||y||ₐ

Question 33

What does the determinant of a matrix have to do with its inverse?

Answer 33

The inverse of a matrix exists if and only if the determinant is non-zero

Question 34

If you have given Γ₂ₓ₂, what is γ₁?

Answer 34

1) the first column vector (₂ₓ₁)

2) called the “first eigenvector”