Math Problems Vault

Problem 3.2.1

(a)(b)

Determine the unknown quantities in the following expressions.

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(a)

$3\mathbf{X} = \begin{bmatrix} 0 & 3 \\ 6 & 9 \end{bmatrix}$ .

(b)

$2\begin{bmatrix} x+2 & y+3 \\ 3 & 0 \end{bmatrix} = \begin{bmatrix} 3 & 6 \\ y & z \end{bmatrix}^{T}$ .

Problem 3.2.2

(a)(b)(c)(d)

Identify each of the following as symmetric, skew symmetric, or neither.

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(a)

$\begin{bmatrix} 1 & -3 & 3 \\ -3 & 4 & -3 \\ 3 & 3 & 0 \end{bmatrix}$ .

(b)

$\begin{bmatrix} 0 & -3 & -3 \\ 3 & 0 & 1 \\ 3 & -1 & 0 \end{bmatrix}$ .

(c)

$\begin{bmatrix} 0 & -3 & -3 \\ -3 & 0 & 3 \\ -3 & 3 & 1 \end{bmatrix}$ .

(d)

$\begin{bmatrix} 1 & 2 & 0 \\ 2 & 1 & 0 \end{bmatrix}$ .

Problem 3.2.3

(a)(b)(c)

Construct an example of a $3 \times 3$ matrix $\mathbf{A}$ that satisfies the following conditions.

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(a)

$\mathbf{A}$ is both symmetric and skew symmetric.

(b)

$\mathbf{A}$ is both hermitian and symmetric.

(c)

$\mathbf{A}$ is skew hermitian.

Problem 3.2.4

Explain why the set of all $n \times n$ symmetric matrices is closed under matrix addition. That is, explain why the sum of two $n \times n$ symmetric matrices is again an $n \times n$ symmetric matrix. Is the set of all $n \times n$ skew-symmetric matrices closed under matrix addition?

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Problem 3.2.5

(a)(b)(c)

Prove that each of the following statements is true.

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(a)

If $\mathbf{A} = [a_{ij}]$ is skew symmetric, then $a_{jj} = 0$ for each $j$ .

(b)

If $\mathbf{A} = [a_{ij}]$ is skew hermitian, then each $a_{jj}$ is a pure imaginary number -- i.e., a multiple of the imaginary unit $\mathrm{i}$ .

(c)

If $\mathbf{A}$ is real and symmetric, then $\mathbf{B} = \mathrm{i}\mathbf{A}$ is skew hermitian.

Problem 3.2.6

(a)(b)

Let $\mathbf{A}$ be any square matrix.

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(a)

Show that $\mathbf{A}+\mathbf{A}^{T}$ is symmetric and $\mathbf{A}-\mathbf{A}^{T}$ is skew symmetric.

(b)

Prove that there is one and only one way to write $\mathbf{A}$ as the sum of a symmetric matrix and a skew-symmetric matrix.

Problem 3.2.7

(a)(b)

If $\mathbf{A}$ and $\mathbf{B}$ are two matrices of the same shape, prove that each of the following statements is true.

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(a)

$(\mathbf{A}+\mathbf{B})^{*} = \mathbf{A}^{*}+\mathbf{B}^{*}$ .

(b)

$(\alpha \mathbf{A})^{*} = \bar{\alpha }\mathbf{A}^{*}$ .

Problem 3.2.8

Recall the spring-system setup of Example 3.2.1: for a line of springs connecting nodes, if node $i$ is displaced $x_i$ units (positive to the left, negative to the right) and consecutive nodes are joined by a spring of stiffness constant $k$ , then the force exerted on an interior node by its two adjacent springs is the sum of $k$ times the relative displacement with each neighbor, and organizing the resulting force balance equations into a linear system produces a symmetric stiffness matrix $\mathbf{K}$ as the coefficient matrix. For two springs joining three nodes with a common stiffness constant $k_1 = k_2 = k$ , this construction gives

\mathbf{K} = k \begin{bmatrix} 1 & -1 & 0 \\ -1 & 2 & -1 \\ 0 & -1 & 1 \end{bmatrix}.

Using the conventions given in Example 3.2.1, determine the stiffness matrix for a system of $n$ identical springs, with stiffness constant $k$ , connected in a line similar to that shown in Figure 3.2.1.

Figure 3.2.1

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