Linear Algebra Practice Set 2

Let $x$ be a non-zero real constant. Let $A = (a_{ij})_{n \times n}$ be an $n$-th order matrix satisfying:

$a_{ii} = 1 + x^2$ for all $i = 1, \ldots, n$

$a_{j,j+1} = x$ for all $j = 1, \ldots, n-1$

$a_{l,l-1} = x$ for all $l = 2, \ldots, n$

All other elements are 0.

Prove that $|A| = \sum_{k=0}^{n} x^{2k}$.

Given vectors in $\mathbb{R}^3$:

(I): $\alpha_1 = (1, -1, 0)^T, \alpha_2 = (-1, 0, 1)^T, \alpha_3 = (1, 2, 2)^T$

(1) Prove that vector group (I) is a basis of $\mathbb{R}^3$;

(2) Use the Gram-Schmidt orthogonalization method to transform basis (I) into an orthonormal basis (II);

(3) Find the transition matrix $M$ from basis (II) to basis (I).

Let $\lambda, \mu$ be real constants. Solve the linear system:

$\begin{cases} x_1 + x_2 + x_3 + x_4 = 1 \\ x_1 + x_2 + \lambda x_3 - x_4 = 1 \\ x_1 + \lambda x_2 + x_3 + \mu x_4 = 1 \\ \lambda x_1 + x_2 + x_3 + x_4 = \mu \end{cases}$

Let $a, b$ be real constants. It is known that matrix

$A = \begin{pmatrix} 1 & -3 & 1 \\ 2 & -3 & 2 \\ -3 & 4 & a \end{pmatrix}$

is similar to matrix $B = \begin{pmatrix} -2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & b \end{pmatrix}$.

(1) Find the values of $a$ and $b$;

(2) Find an invertible matrix $P$ such that $P^{-1}AP = B$.

Let $m, n, p$ be positive integers. Let $A$ be an $m \times n$ real matrix and $B$ be a $p \times n$ real matrix.

Define $W = \{AX \in \mathbb{R}^m : BX = 0_p\}$, where $0_p$ is the $p$-dimensional zero column vector.

(1) Prove that $W$ is a subspace of $\mathbb{R}^m$;

(2) If $r\left(\begin{pmatrix} A \\ B \end{pmatrix}\right) = r_1$ and $r(B) = r_2$, find $\dim(W)$.

Given the real quadratic form:

$f(x_1, x_2, x_3) = 2x_1x_2 - 6x_1x_3 + 2x_2x_3$

Find the canonical form of this quadratic form, and determine its rank and positive index of inertia.

(1) State the Frobenius inequality for ranks of three $n$-th order matrices $A, B, C$;

(2) Let $A$ be an $n$-th order real matrix. Prove that:

① $2r(A^2) \leq r(A^3) + r(A)$

② $2021 \cdot r(A^2) \leq r(A^{2022}) + 2020 \cdot r(A)$

(1) Let $A, B$ be $n$-th order real symmetric matrices with $A$ positive definite. Prove that $AB$ is diagonalizable;

(2) Let $D$ be an $n$-th order real orthogonal matrix. Given that $D + E$ is invertible, prove that:

① The matrix equation $D(E + X) = E - X$ has a solution $X$

② The solution $X$ in ① is a real skew-symmetric matrix