MathDB

AGMC 2021 Prelim Q4

Source:

January 10, 2023

linear algebramatrixcomplex numbersvector

Problem Statement

Let

n

be a positive integer. For any positive integer

k

, let

0_k=diag\{\underbrace{0, ...,0}_{k}\}

be a

k \times k

zero matrix. Let

Y=\begin{pmatrix} 0_n & A \\ A^t & 0_{n+1} \end{pmatrix}

be a

(2n+1) \times (2n+1)

where

A=(x_{i, j})_{1\leq i \leq n, 1\leq j \leq n+1}

is a

n \times (n+1)

real matrix. Let

A^T

be transpose matrix of

A

i.e.

(n+1) \times n

matrix, the element of

(j, i)

is

x_{i, j}

. (a) Let complex number

\lambda

be an eigenvalue of

k \times k

matrix

X

. If there exists nonzero column vectors

v=(x_1, ..., x_k)^t

such that

Xv=\lambda v

. Prove that 0 is the eigenvalue of

Y

and the other eigenvalues of

Y

can be expressed as a form of

\pm \sqrt{\lambda}

where nonnegative real number

\lambda

is the eigenvalue of

AA^t

. (b) Let

n=3

and

a_1

,

a_2

,

a_3

,

a_4

are

4

distinct positive real numbers. Let

a=\sqrt[]{\sum_{1\leq i \leq 4}^{}a^{2}_{i}}

and

x_{i,j}=a_i\delta_{i,j}+a_j\delta_{4,j}-\frac{1}{a^2}(a^2_{i}+a^2_{4})a_j

where

1\leq i \leq 3, 1\leq j \leq 4

,

\delta_{i, j}= \begin{cases} 1 \text{ if } i=j\\ 0 \text{ if } i\neq j\\ \end{cases}\,

. Prove that

Y

has 7 distinct eigenvalue.

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