Math Problems Vault

Problem 8.1

(a)(b)

Let $A$ represent an $m \times n$ matrix. Show that

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(a)

if $A$ has a right inverse, then $n \geq m$ and

(b)

if $A$ has a left inverse, then $m \geq n$ .

Problem 8.2

(a)(b)

An $n \times n$ matrix $A$ is said to be involutory if $A^2 = I$ , that is, if $A$ is invertible and is its own inverse.

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(a)

Show that an $n \times n$ matrix $A$ is involutory if and only if $(I-A)(I+A) = 0$ .

(b)

Show that a $2 \times 2$ matrix $A = \left(\begin{smallmatrix} a & b \\ c & d \end{smallmatrix}\right)$ is involutory if and only if (1) $a^2+bc = 1$ and $d = -a$ or (2) $b = c = 0$ and $d = a = \pm 1$ .

Problem 8.3

Let $A$ represent an $n \times n$ nonnull symmetric matrix, and let $B$ represent an $n \times r$ matrix of full column rank $r$ and $T$ an $r \times n$ matrix of full row rank $r$ such that $A = BT$ . Show that the $r \times r$ matrix $TB$ is nonsingular.

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Remark.

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Observe that $A'A = A^2 = BTBT$ .

Problem 8.4

(a)(b)

Let $A$ represent an $n \times n$ matrix, and partition $A$ as $A = (A_1, A_2)$ .

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(a)

Show that if $A$ is invertible and $A^{-1}$ is partitioned as $A^{-1} = \left(\begin{smallmatrix} B_1 \\ B_2 \end{smallmatrix}\right)$ (where $B_1$ has the same number of rows as $A_1$ has columns), then

\begin{align} & B_1 A_1 = I, \quad B_1 A_2 = 0, \quad B_2 A_1 = 0, \quad B_2 A_2 = I, \tag {E.1} \\ & A_1 B_1 = I - A_2 B_2, \quad A_2 B_2 = I - A_1 B_1. \tag {E.2} \end{align}

(b)

Show that if $A$ is orthogonal, then

\begin{align} & A_1' A_1 = I, \quad A_1' A_2 = 0, \quad A_2' A_1 = 0, \quad A_2' A_2 = I, \tag {E.3} \\ & A_1 A_1' = I - A_2 A_2', \quad A_2 A_2' = I - A_1 A_1'. \tag {E.4} \end{align}

Problem 8.5

Let $A$ represent an $m \times n$ nonnull matrix of rank $r$ . Show that there exists an $m \times m$ orthogonal matrix whose first $r$ columns span $\mathcal{C}(A)$ .

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Problem 8.6

Let $T$ represent an $n \times n$ triangular matrix. Show that $\mathrm{rank}(T)$ is greater than or equal to the number of nonzero diagonal elements in $T$ .

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Problem 8.7

(a)(b)

Let

A = \left(\begin{smallmatrix} A_{11} & A_{12} & \cdots & A_{1r} \\ 0 & A_{22} & \cdots & A_{2r} \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & A_{rr} \end{smallmatrix}\right), \qquad B = \left(\begin{smallmatrix} B_{11} & 0 & \cdots & 0 \\ B_{21} & B_{22} & & 0 \\ \vdots & \vdots & \ddots & \\ B_{r1} & B_{r2} & \cdots & B_{rr} \end{smallmatrix}\right)

represent respectively an $n \times n$ upper block-triangular matrix whose $ij$ th block $A_{ij}$ is of dimensions $n_i \times n_j$ ( $j \geq i = 1, \ldots , r$ ) and an $n \times n$ lower block-triangular matrix whose $ij$ th block $B_{ij}$ is of dimensions $n_i \times n_j$ ( $j \leq i = 1, \ldots , r$ ).

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(a)

Assume that $A$ and $B$ are invertible, and ``recall'' that

A^{-1} = \left(\begin{smallmatrix} F_{11} & F_{12} & \cdots & F_{1r} \\ 0 & F_{22} & \cdots & F_{2r} \\ \vdots & & \ddots & \vdots \\ 0 & 0 & \cdots & F_{rr} \end{smallmatrix}\right), \qquad B^{-1} = \left(\begin{smallmatrix} G_{11} & 0 & \cdots & 0 \\ G_{21} & G_{22} & & 0 \\ \vdots & \vdots & \ddots & \\ G_{r1} & G_{r2} & \cdots & G_{rr} \end{smallmatrix}\right),

where

\begin{align} F_{ii} & = A_{ii}^{-1}, \quad F_{ij} = -A_{ii}^{-1} \sum _{k=i+1}^{j} A_{ik} F_{kj} \quad (j > i = 1, \ldots , r), \tag {*} \\ G_{ii} & = B_{ii}^{-1}, \quad G_{ij} = -B_{ii}^{-1} \sum _{k=j}^{i-1} B_{ik} G_{kj} \quad (j < i = 1, \ldots , r). \tag {**} \end{align}

Show that the submatrices $F_{ij}$ ( $j \geq i = 1, \ldots , r$ ) and $G_{ij}$ ( $j \leq i = 1, \ldots , r$ ) are also expressible as

\begin{align} F_{jj} & = A_{jj}^{-1}, \quad F_{ij} = -\left( \sum _{k=i}^{j-1} F_{ik} A_{kj} \right) A_{jj}^{-1} \quad (i < j = 1, \ldots , r), \tag {E.5} \\ G_{jj} & = B_{jj}^{-1}, \quad G_{ij} = -\left( \sum _{k=j+1}^{i} G_{ik} B_{kj} \right) B_{jj}^{-1} \quad (i > j = 1, \ldots , r). \tag {E.6} \end{align}

Do so by applying results (**) and (*) to $A'$ and $B'$ , respectively.

(b)

Formulas () form the basis for an algorithm for computing $A^{-1}$ in $r$ steps: the first step is to compute the matrix $F_{rr} = A_{rr}^{-1}$ ; the $(r-i+1)$ th step is to compute the matrices $F_{ii}, F_{i,i+1}, \ldots , F_{ir}$ from formulas () ( $i = r-1, r-2, \ldots , 1$ ). Similarly, formulas () form the basis for an algorithm for computing $B^{-1}$ in $r$ steps: the first step is to compute the matrix $G_{11} = B_{11}^{-1}$ ; the $i$ th step is to compute the matrices $G_{i1}, G_{i2}, \ldots , G_{ii}$ from formulas () ( $i = 2, \ldots , r$ ). Describe how formulas (E.5) and (E.6) in Part (a) can be used to devise $r$ -step algorithms for computing $A^{-1}$ and $B^{-1}$ , and indicate how these algorithms differ from those based on formulas (*) and (**).