(a) [1] 矩阵非奇异是什么意思？
(b) [1] 如果给你一个矩阵，你如何检查它是非奇异的？
(c) [1] 举一个只有 f0 的 3×3 矩阵的例子； 1g-条目是单数。
(d) [1] 举一个只有 f0 的 3×3 矩阵的例子； 1g 条目是非奇异的。
(e) [1] 比较这两个矩阵的特征值。你能说什么

(f) [3] 证明 Z (X )W 1 具有 N(0; I) 分布，其中 I 是 p p 恒等式

(a) [1] 来自带参数的多元正态分布的样本 n = 1 个实例

(b) [1] 计算的特征值并将它们绘制为线图。在什么值下

Question 1

Let X be a p-dimensional random vector with multivariate Normal distribution X  N(; V )
where  is a p-dimensional vector and V is a non-singular p  p symmetric matrix.

(a) [1] What does it mean for a matrix to be non-singular?
(b) [1] If you are given a matrix how could you check that it is non-singular?
(c) [1] Come up with an example of a 3×3 matrix with only f0; 1g-entries that is singular.
(d) [1] Come up with an example of a 3×3 matrix with only f0; 1g-entries that is non-singular.
(e) [1] Compare the eigenvalues of these two matrices. What can you say about the
eigenvalues of the singular matrix?
(f) [3] Show that Z  (X )W 1 has a N(0; I) distribution where I is a p  p identity
matrix and W is a symmetric matrix satisfying W2 = V .

Question 2

Consider the p = 768 dimensional vector  and p  p covariance matrix  contained in
the ﬁles ‘mu.txt’ and ‘Sigma.txt’. Read the values from these ﬁles into R.

(a) [1] Sample n = 1 instance from a multivariate Normal distribution with parameters 
and . Reshape the p = 768 dimensional sample to a matrix of size r c = 2828
and plot the matrix. Can you guess what random images you are generating? You
may need to sample/try this a couple of times to guess correctly.

(b) [1] Calculate the eigenvalues of  and plot them as a line plot. At what value do the
eigenvalues seem to level oﬀ at? i.e., what is the value where the “elbow” appears.

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