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MH4501 Assignment

创建日期：2024-06-11 15:40:52

所属分类：数学mathematics

标签： MH4501

Multivariate Analysis

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MH4501 Multivariate Analysis

Assignment
Question 1
Let A, B and C be n× n square matrices, with C non-singular, such that A = CBC−1 holds.
1. Express B10 as a function of A, simplifying as much as possible.
2. In general, do A and B have the same eigenvalues? The same eigenvectors?
Question 2
1. Let A be an n × n matrix. Show that the quadratic form of A is equal to the one of the matrix
S = 12 (A+A
⊤)
2. Let Σ be a n× n symmetric matrix.
(a) Suppose that Σ is a covariance matrix. Let Z = (Z1, . . . , Zn)
⊤ be a random vector with covariance
matrix Σ and let Y =
∑n
i=1 xiZi with xi ∈ R for all i ∈ {1, . . . , n}. Prove that Σ is positive semi-
definite using the fact that var(Y ) ≥ 0.
(b) Suppose that Σ is positive semi-definite. Let Z ′ = (Z ′1, . . . , Z
′
n)
⊤ be a n× 1 random vector with
mean 0 and covariance I, and let
Zi =
n∑
k=1
Uik
√
ΛkkZ
′
k, i = 1, . . . , n,
with U and Λ an eigenvalue decomposition of Σ, i.e. Σ = UΛU⊤. Show that the covariance of
Z = (Z1, . . . , Zn)
⊤ is Σ.
(c) What can you deduce from 2.(a) and 2.(b) regarding covariance matrices and positive semi-
definiteness?
3. Let Z = (Z1, . . . , Zn)
⊤ be an n × 1 random vector with a covariance matrix Σ that is positive semi-
definite but not positive definite. Show that there exist xi ∈ R, i ∈ {1, . . . , n}, such that
var
( n∑
i=1
xiZi
)
= 0.
1
Question 3
Consider 100 samples X1, . . . , X100 from the distribution N8(µ,Σ). For each of the following questions,
justify your answer.
1. What is the distribution of the sample mean X¯?
2. What is the distribution of
√
n(X¯ − µ)?
3. Let S be the sample covariance matrix of X1, . . . , X100, what is the distribution of (n− 1)S?
Question 4
Consider a 3× 1 normal random vector Z = (Z1, Z2, Z3)⊤ with mean (56, 2, 34)⊤ and covariance matrix2 0 10 3 0
1 0 4
 .
1. Are Z2 and (Z1, Z3) independent?
2. Are Z1 + 3Z2 − 2Z3 and Z1 independent?

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