Week 1

Question 1

What is the new Multivariate Linear Regression formula?

Answer 1

Question 2

What is A’?

Answer 2

It is the Transpose of vector A

Question 3

When has a matrix full rank?

Answer 3

If AX = I_n, so then X = A^-1

Question 4

When is a square matrix idempotent?

Answer 4

If AA = A

Question 5

When is a square matrix symmetric?

Answer 5

If A’ = A

Question 6

When is a matrix orthogonal?

Answer 6

If A’A = I_n

Question 7

When are two matrices orthogonal to each other?

Answer 7

If A’B = 0

Question 8

What is the trace of a matrix?

Answer 8

The sum of its diagnoal elements

Question 9

What are the 3 important properties of a trace?

Answer 9

1. tr(𝛼A) = 𝛼 tr(A)
2. tr(A’) = tr(A)
3. tr(B’C) = tr(CB’)

Question 10

How is the rank determined?

Answer 10

Column rank: number of independent columns

Row rank: number of independent rows

Question 11

What holds for an idempotent matrix?

Answer 11

r(A) = tr(A)

Question 12

How to calculate the multivariate expectation?

Answer 12

Question 13

What is the definition of the multivariate variance?

Answer 13

Question 14

How to calculate the multivariate variance?

Answer 14

Var(X) = E[(x-E(x))(x- E(x))’]

Question 15

How to calculate the multivariate covariance?

Answer 15

Cov(x, y) = E[(x - E(x))(y - E(y))’]

Question 16

What are the rules for the covariance and variance with matrices in front of the variables?

Answer 16

Cov(Ax, By) = A Cov(x, y) B’

Var(x) = A Var(x) A’