Math Problems Vault

Problem 17.1

(a)(b)

Let $\mathcal{U}$ and $\mathcal{V}$ represent subspaces of $\mathbb {R}^{m \times n}$ .

?

(a)

Show that $\mathcal{U} \cup \mathcal{V} \subset \mathcal{U} + \mathcal{V}$ .

(b)

Show that $\mathcal{U} + \mathcal{V}$ is the smallest subspace (of $\mathbb {R}^{m \times n}$ ) that contains $\mathcal{U} \cup \mathcal{V}$ , or, equivalently [in light of Part (a)], show that, for any subspace $\mathcal{W}$ such that $\mathcal{U} \cup \mathcal{V} \subset \mathcal{W}$ , $\mathcal{U} + \mathcal{V} \subset \mathcal{W}$ .

Problem 17.2

(a)(b)(c)

Let $\mathbf{A} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \\ 0 & 0 \end{pmatrix}$ and $\mathbf{B} = \begin{pmatrix} 0 & 2 \\ 1 & 1 \\ 2 & 3 \end{pmatrix}$ . Find

?

(a)

a basis for $C(\mathbf{A}) + C(\mathbf{B})$ ,

(b)

a basis for $C(\mathbf{A}) \cap C(\mathbf{B})$ , and

(c)

a vector in $C(\mathbf{A}) + C(\mathbf{B})$ that is not in $C(\mathbf{A}) \cup C(\mathbf{B})$ .

Problem 17.3

(a)(b)

Let $\mathcal{U}$ , $\mathcal{W}$ , and $\mathcal{X}$ represent subspaces of a linear space $\mathcal{V}$ of matrices, and let $\mathbf{Y}$ represent an arbitrary matrix in $\mathcal{V}$ .

?

(a)

Show (1) that if $\mathbf{Y} \perp \mathcal{W}$ and $\mathbf{Y} \perp \mathcal{X}$ , then $\mathbf{Y} \perp (\mathcal{W} + \mathcal{X})$ , and (2) that if $\mathcal{U} \perp \mathcal{W}$ and $\mathcal{U} \perp \mathcal{X}$ , then $\mathcal{U} \perp (\mathcal{W} + \mathcal{X})$ .

(b)

Show (1) that $(\mathcal{U} + \mathcal{W})^{\perp } = \mathcal{U}^{\perp } \cap \mathcal{W}^{\perp }$ and (2) that $(\mathcal{U} \cap \mathcal{W})^{\perp } = \mathcal{U}^{\perp } + \mathcal{W}^{\perp }$ .

Problem 17.4

(a)(b)(c)

Let $\mathcal{U}$ , $\mathcal{W}$ , and $\mathcal{X}$ represent subspaces of a linear space $\mathcal{V}$ of matrices.

?

(a)

Show that $(\mathcal{U} \cap \mathcal{W}) + (\mathcal{U} \cap \mathcal{X}) \subset \mathcal{U} \cap (\mathcal{W} + \mathcal{X})$ .

(b)

Show (via an example) that $\mathcal{U} \cap \mathcal{W} = \left\{ \mathbf{0}\right\}$ and $\mathcal{U} \cap \mathcal{X} = \left\{ \mathbf{0}\right\}$ does not necessarily imply that $\mathcal{U} \cap (\mathcal{W} + \mathcal{X}) = \left\{ \mathbf{0}\right\}$ .

(c)

Show that if $\mathcal{W} \subset \mathcal{U}$ , then (1) $\mathcal{U} + \mathcal{W} = \mathcal{U}$ and (2) $\mathcal{U} \cap (\mathcal{W} + \mathcal{X}) = \mathcal{W} + (\mathcal{U} \cap \mathcal{X})$ .

Problem 17.5

Let $\mathcal{U}_{1}, \mathcal{U}_{2}, \ldots , \mathcal{U}_{k}$ represent subspaces of $\mathbb {R}^{m \times n}$ . Show that if, for $j = 1, 2, \ldots , k$ , $\mathcal{U}_{j}$ is spanned by a (finite nonempty) set of ( $m \times n$ ) matrices $\mathbf{U}_{1}^{(j)}, \ldots , \mathbf{U}_{r_{j}}^{(j)}$ , then

\mathcal{U}_{1} + \mathcal{U}_{2} + \cdots + \mathcal{U}_{k} = \operatorname {sp}(\mathbf{U}_{1}^{(1)}, \ldots , \mathbf{U}_{r_{1}}^{(1)}, \mathbf{U}_{1}^{(2)}, \ldots , \mathbf{U}_{r_{2}}^{(2)}, \ldots , \mathbf{U}_{1}^{(k)}, \ldots , \mathbf{U}_{r_{k}}^{(k)}).

?

Problem 17.6

Let $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ represent subspaces of $\mathbb {R}^{m \times n}$ . The $k$ subspaces $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are said to be independent if, for matrices $\mathbf{U}_{1} \in \mathcal{U}_{1}, \ldots , \mathbf{U}_{k} \in \mathcal{U}_{k}$ , the only solution to the matrix equation

\mathbf{U}_{1} + \cdots + \mathbf{U}_{k} = \mathbf{0} \tag {E.1}

is $\mathbf{U}_{1} = \cdots = \mathbf{U}_{k} = \mathbf{0}$ .

?

(a)

Show that $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are independent if and only if, for $i = 2, \ldots , k$ , $\mathcal{U}_{i}$ and $\mathcal{U}_{1} + \cdots + \mathcal{U}_{i-1}$ are essentially disjoint.

(b)

Show that $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are independent if and only if, for $i = 1, \ldots , k$ , $\mathcal{U}_{i}$ and $\mathcal{U}_{1} + \cdots + \mathcal{U}_{i-1} + \mathcal{U}_{i+1} + \cdots + \mathcal{U}_{k}$ are essentially disjoint.

(c)

Use the results of Exercise 3 [along with Part (a) or (b)] to show that if $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are (pairwise) orthogonal, then they are independent.

(d)

Assuming that $\mathcal{U}_{1}, \mathcal{U}_{2}, \ldots , \mathcal{U}_{k}$ are of dimension one or more and letting $\left\{ \mathbf{U}_{1}^{(j)}, \ldots , \mathbf{U}_{r_{j}}^{(j)}\right\}$ represent any linearly independent set of matrices in $\mathcal{U}_{j}$ ( $j = 1, 2, \ldots , k$ ), show that if $\mathcal{U}_{1}, \mathcal{U}_{2}, \ldots , \mathcal{U}_{k}$ are independent, then the combined set $\left\{ \mathbf{U}_{1}^{(1)}, \ldots , \mathbf{U}_{r_{1}}^{(1)}, \mathbf{U}_{1}^{(2)}, \ldots , \mathbf{U}_{r_{2}}^{(2)}, \ldots , \mathbf{U}_{1}^{(k)}, \ldots , \mathbf{U}_{r_{k}}^{(k)}\right\}$ is linearly independent.

(e)

Assuming that $\mathcal{U}_{1}, \mathcal{U}_{2}, \ldots , \mathcal{U}_{k}$ are of dimension one or more, show that $\mathcal{U}_{1}, \mathcal{U}_{2}, \ldots , \mathcal{U}_{k}$ are independent if and only if, for every nonnull matrix $\mathbf{U}_{1}$ in $\mathcal{U}_{1}$ , every nonnull matrix $\mathbf{U}_{2}$ in $\mathcal{U}_{2}$ , , and every nonnull matrix $\mathbf{U}_{k}$ in $\mathcal{U}_{k}$ , $\mathbf{U}_{1}, \mathbf{U}_{2}, \ldots , \mathbf{U}_{k}$ are linearly independent.

(f)

For $j = 1, \ldots , k$ , let $p_{j} = \dim (\mathcal{U}_{j})$ , and let $S_{j}$ represent a basis for $\mathcal{U}_{j}$ ( $j = 1, \ldots , k$ ). Define $S$ to be the set of $\sum_{j=1}^{k} p_{j}$ matrices obtained by combining all of the matrices in $S_{1}, \ldots , S_{k}$ into a single set. Use the result of Exercise 5 [along with Part (d)] to show that (1) if $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are independent, then $S$ is a basis for $\mathcal{U}_{1} + \cdots + \mathcal{U}_{k}$ ; and (2) if $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are not independent, then $S$ contains a proper subset that is a basis for $\mathcal{U}_{1} + \cdots + \mathcal{U}_{k}$ .

(g)

Show that (1) if $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are independent, then

\dim (\mathcal{U}_{1} + \cdots + \mathcal{U}_{k}) = \dim (\mathcal{U}_{1}) + \cdots + \dim (\mathcal{U}_{k});

and (2) if $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are not independent, then

\dim (\mathcal{U}_{1} + \cdots + \mathcal{U}_{k}) < \dim (\mathcal{U}_{1}) + \cdots + \dim (\mathcal{U}_{k}).

Problem 17.7

(a)(b)

Let $\mathbf{A}_{1}, \ldots , \mathbf{A}_{k}$ represent matrices having the same number of rows, and let $\mathbf{B}_{1}, \ldots , \mathbf{B}_{k}$ represent matrices having the same number of columns. Adopting the terminology of Exercise 6, use Part (g) of that exercise to show

?

(a)

that if $C(\mathbf{A}_{1}), \ldots , C(\mathbf{A}_{k})$ are independent, then

\operatorname {rank}(\mathbf{A}_{1}, \ldots , \mathbf{A}_{k}) = \operatorname {rank}(\mathbf{A}_{1}) + \cdots + \operatorname {rank}(\mathbf{A}_{k}),

and if $C(\mathbf{A}_{1}), \ldots , C(\mathbf{A}_{k})$ are not independent, then

\operatorname {rank}(\mathbf{A}_{1}, \ldots , \mathbf{A}_{k}) < \operatorname {rank}(\mathbf{A}_{1}) + \cdots + \operatorname {rank}(\mathbf{A}_{k})

and

(b)

that if $\mathcal{R}(\mathbf{B}_{1}), \ldots , \mathcal{R}(\mathbf{B}_{k})$ are independent, then

\operatorname {rank}\begin{pmatrix} \mathbf{B}_{1} \\ \vdots \\ \mathbf{B}_{k} \end{pmatrix} = \operatorname {rank}(\mathbf{B}_{1}) + \cdots + \operatorname {rank}(\mathbf{B}_{k}),

and if $\mathcal{R}(\mathbf{B}_{1}), \ldots , \mathcal{R}(\mathbf{B}_{k})$ are not independent, then

\operatorname {rank}\begin{pmatrix} \mathbf{B}_{1} \\ \vdots \\ \mathbf{B}_{k} \end{pmatrix} < \operatorname {rank}(\mathbf{B}_{1}) + \cdots + \operatorname {rank}(\mathbf{B}_{k}).

Problem 17.8

(a)(b)

Letting $\mathbf{A}$ represent an $m \times n$ matrix and $\mathbf{B}$ an $m \times p$ matrix, show, by for instance using the result of Part (c)-(2) of Exercise 4 in combination with the result

C(\mathbf{A}, \mathbf{B}) = C[\mathbf{A}, (\mathbf{I} - \mathbf{A}\mathbf{A}^{-})\mathbf{B}] = C(\mathbf{A}) \oplus C[(\mathbf{I} - \mathbf{A}\mathbf{A}^{-})\mathbf{B}], \tag {*}

that

?

(a)

$C[(\mathbf{I} - \mathbf{A}\mathbf{A}^{-})\mathbf{B}] = C(\mathbf{I} - \mathbf{A}\mathbf{A}^{-}) \cap C(\mathbf{A}, \mathbf{B})$ and

(b)

$C[(\mathbf{I} - \mathbf{P}_{\mathbf{A}})\mathbf{B}] = \mathcal{N}(\mathbf{A}') \cap C(\mathbf{A}, \mathbf{B})$ .

Problem 17.9

Let $\mathbf{A} = (\mathbf{T}, \mathbf{U})$ and $\mathbf{B} = (\mathbf{V}, \mathbf{0})$ , where $\mathbf{T}$ is an $m \times p$ matrix, $\mathbf{U}$ an $m \times q$ matrix, and $\mathbf{V}$ an $n \times p$ matrix, and suppose that $\mathbf{U}$ is of full row rank. Show that $\mathcal{R}(\mathbf{A})$ and $\mathcal{R}(\mathbf{B})$ are essentially disjoint [even if $\mathcal{R}(\mathbf{T})$ and $\mathcal{R}(\mathbf{V})$ are not essentially disjoint].

?

Problem 17.10

(a)(b)

To what extent does the formula

\operatorname {rank}\begin{pmatrix} \mathbf{T} & \mathbf{U} \\ \mathbf{V} & \mathbf{0} \end{pmatrix} = \operatorname {rank}(\mathbf{U}) + \operatorname {rank}(\mathbf{V}) + \operatorname {rank}[(\mathbf{I} - \mathbf{U}\mathbf{U}^{-})\mathbf{T}(\mathbf{I} - \mathbf{V}^{-}\mathbf{V})] \tag {*}

[where $\mathbf{T}$ is an $m \times p$ matrix, $\mathbf{U}$ an $m \times q$ matrix, and $\mathbf{V}$ an $n \times p$ matrix] simplify in

?

(a)

the special case where $C(\mathbf{T})$ and $C(\mathbf{U})$ are essentially disjoint [but $\mathcal{R}(\mathbf{T})$ and $\mathcal{R}(\mathbf{V})$ are not necessarily essentially disjoint] and

(b)

the special case where $\mathcal{R}(\mathbf{T})$ and $\mathcal{R}(\mathbf{V})$ are essentially disjoint.

Problem 17.11

(a)(b)

Let $\mathbf{T}$ represent an $m \times p$ matrix, $\mathbf{U}$ an $m \times q$ matrix, and $\mathbf{V}$ an $n \times p$ matrix. Further, define $\mathbf{E}_{\mathbf{T}} = \mathbf{I} - \mathbf{T}\mathbf{T}^{-}$ , $\mathbf{F}_{\mathbf{T}} = \mathbf{I} - \mathbf{T}^{-}\mathbf{T}$ , $\mathbf{X} = \mathbf{E}_{\mathbf{T}}\mathbf{U}$ , and $\mathbf{Y} = \mathbf{V}\mathbf{F}_{\mathbf{T}}$ . Show

?

(a)

that the partitioned matrix $\begin{pmatrix} \mathbf{T}^{-} - \mathbf{T}^{-}\mathbf{U}\mathbf{X}^{-}\mathbf{E}_{\mathbf{T}} \\ \mathbf{X}^{-}\mathbf{E}_{\mathbf{T}} \end{pmatrix}$ is a generalized inverse of the partitioned matrix $(\mathbf{T}, \mathbf{U})$ and

(b)

that the partitioned matrix $(\mathbf{T}^{-} - \mathbf{F}_{\mathbf{T}}\mathbf{Y}^{-}\mathbf{V}\mathbf{T}^{-},\ \mathbf{F}_{\mathbf{T}}\mathbf{Y}^{-})$ is a generalized inverse of the partitioned matrix $\begin{pmatrix} \mathbf{T} \\ \mathbf{V} \end{pmatrix}$ .

Do so by applying formula (E.1) from Part (a) of Exercise 10.10 to the partitioned matrices $\begin{pmatrix} \mathbf{T} & \mathbf{U} \\ \mathbf{0} & \mathbf{0} \end{pmatrix}$ and $\begin{pmatrix} \mathbf{T} & \mathbf{0} \\ \mathbf{V} & \mathbf{0} \end{pmatrix}$ and by making use of the result that for any generalized inverse $\mathbf{G} = \begin{pmatrix} \mathbf{G}_{1} \\ \mathbf{G}_{2} \end{pmatrix}$ of the partitioned matrix $(\mathbf{A}, \mathbf{B})$ and any generalized inverse $\mathbf{H} = (\mathbf{H}_{1}, \mathbf{H}_{2})$ of the partitioned matrix $\begin{pmatrix} \mathbf{A} \\ \mathbf{C} \end{pmatrix}$ (where $\mathbf{A}$ is an $m \times n$ matrix, $\mathbf{B}$ an $m \times p$ matrix, and $\mathbf{C}$ a $q \times n$ matrix and where $\mathbf{G}_{1}$ has $n$ rows and $\mathbf{H}_{1}$ $m$ columns), (1) $\mathbf{G}_{1}$ is a generalized inverse of $\mathbf{A}$ and $\mathbf{G}_{2}$ a generalized inverse of $\mathbf{B}$ if and only if $C(\mathbf{A})$ and $C(\mathbf{B})$ are essentially disjoint, and, similarly, (2) $\mathbf{H}_{1}$ is a generalized inverse of $\mathbf{A}$ and $\mathbf{H}_{2}$ a generalized inverse of $\mathbf{C}$ if and only if $\mathcal{R}(\mathbf{A})$ and $\mathcal{R}(\mathbf{C})$ are essentially disjoint.

Problem 17.12

(a)(b)

Let $\mathbf{T}$ represent an $m \times p$ matrix, $\mathbf{U}$ an $m \times q$ matrix, and $\mathbf{V}$ an $n \times p$ matrix. And, let $\begin{pmatrix} \mathbf{G}_{11} & \mathbf{G}_{12} \\ \mathbf{G}_{21} & \mathbf{G}_{22} \end{pmatrix}$ (where $\mathbf{G}_{11}$ is of dimensions $p \times m$ ) represent a generalized inverse of the partitioned matrix $\begin{pmatrix} \mathbf{T} & \mathbf{U} \\ \mathbf{V} & \mathbf{0} \end{pmatrix}$ . Show that

?

(a)

if $\mathbf{G}_{11}$ is a generalized inverse of $\mathbf{T}$ and $\mathbf{G}_{12}$ a generalized inverse of $\mathbf{V}$ , then $\mathcal{R}(\mathbf{T})$ and $\mathcal{R}(\mathbf{V})$ are essentially disjoint, and

(b)

if $\mathbf{G}_{11}$ is a generalized inverse of $\mathbf{T}$ and $\mathbf{G}_{21}$ a generalized inverse of $\mathbf{U}$ , then $C(\mathbf{T})$ and $C(\mathbf{U})$ are essentially disjoint.

Problem 17.13

(a)(b)

?

(a)

Generalize the result that, for any two subspaces $\mathcal{U}$ and $\mathcal{V}$ of $\mathbb {R}^{m \times n}$ ,

\dim (\mathcal{U} + \mathcal{V}) = \dim (\mathcal{U}) + \dim (\mathcal{V}) - \dim (\mathcal{U} \cap \mathcal{V}), \tag {*}

Do so by showing that, for any $k$ subspaces $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ ,

\dim (\mathcal{U}_{1} + \cdots + \mathcal{U}_{k}) = \dim (\mathcal{U}_{1}) + \cdots + \dim (\mathcal{U}_{k}) - \sum _{i=2}^{k} \dim [(\mathcal{U}_{1} + \cdots + \mathcal{U}_{i-1}) \cap \mathcal{U}_{i}]. \tag {E.2}

(b)

Generalize the result that, for any $m \times n$ matrix $\mathbf{A}$ , $m \times p$ matrix $\mathbf{B}$ , and $q \times n$ matrix $\mathbf{C}$ ,

\operatorname {rank}(\mathbf{A}, \mathbf{B}) = \operatorname {rank}(\mathbf{A}) + \operatorname {rank}(\mathbf{B}) - \dim [C(\mathbf{A}) \cap C(\mathbf{B})],

\operatorname {rank}\begin{pmatrix} \mathbf{A} \\ \mathbf{C} \end{pmatrix} = \operatorname {rank}(\mathbf{A}) + \operatorname {rank}(\mathbf{C}) - \dim [\mathcal{R}(\mathbf{A}) \cap \mathcal{R}(\mathbf{C})].

Do so by showing that, for any matrices $\mathbf{A}_{1}, \ldots , \mathbf{A}_{k}$ having the same number of rows,

\operatorname {rank}(\mathbf{A}_{1}, \ldots , \mathbf{A}_{k}) = \operatorname {rank}(\mathbf{A}_{1}) + \cdots + \operatorname {rank}(\mathbf{A}_{k}) - \sum _{i=2}^{k} \dim [C(\mathbf{A}_{1}, \ldots \mathbf{A}_{i-1}) \cap C(\mathbf{A}_{i})]

and, for any matrices $\mathbf{B}_{1}, \ldots , \mathbf{B}_{k}$ having the same number of columns,

\operatorname {rank}\begin{pmatrix} \mathbf{B}_{1} \\ \vdots \\ \mathbf{B}_{k} \end{pmatrix} = \operatorname {rank}(\mathbf{B}_{1}) + \cdots + \operatorname {rank}(\mathbf{B}_{k}) - \sum _{i=2}^{k} \dim \left[\mathcal{R}\begin{pmatrix} \mathbf{B}_{1} \\ \vdots \\ \mathbf{B}_{i-1} \end{pmatrix} \cap \mathcal{R}(\mathbf{B}_{i})\right].

Problem 17.14

Show that, for any $m \times n$ matrix $\mathbf{A}$ , $n \times q$ matrix $\mathbf{C}$ , and $q \times p$ matrix $\mathbf{B}$ ,

\operatorname {rank}\left\{ [\mathbf{I} - \mathbf{C}\mathbf{B}(\mathbf{C}\mathbf{B})^{-}]\mathbf{C}[\mathbf{I} - (\mathbf{A}\mathbf{C})^{-}\mathbf{A}\mathbf{C}]\right\} = \operatorname {rank}(\mathbf{A}) + \operatorname {rank}(\mathbf{C}) - \operatorname {rank}(\mathbf{A}\mathbf{C}) - n + \operatorname {rank}\left\{ [\mathbf{I} - \mathbf{C}\mathbf{B}(\mathbf{C}\mathbf{B})^{-}](\mathbf{I} - \mathbf{A}^{-}\mathbf{A})\right\} .

?

Remark.

?

Apply the equality

\operatorname {rank}(\mathbf{A}\mathbf{C}) = \operatorname {rank}(\mathbf{A}) + \operatorname {rank}(\mathbf{C}) - n + \operatorname {rank}[(\mathbf{I} - \mathbf{C}\mathbf{C}^{-})(\mathbf{I} - \mathbf{A}^{-}\mathbf{A})] \tag {*}

to the product $\mathbf{A}(\mathbf{C}\mathbf{B})$ , and make use of the equality

\operatorname {rank}(\mathbf{A}\mathbf{C}\mathbf{B}) = \operatorname {rank}(\mathbf{A}\mathbf{C}) + \operatorname {rank}(\mathbf{C}\mathbf{B}) - \operatorname {rank}(\mathbf{C}) + \operatorname {rank}\left\{ [\mathbf{I} - \mathbf{C}\mathbf{B}(\mathbf{C}\mathbf{B})^{-}]\mathbf{C}[\mathbf{I} - (\mathbf{A}\mathbf{C})^{-}\mathbf{A}\mathbf{C}]\right\} . \tag {**}

Problem 17.15

Show that if an $n \times n$ matrix $\mathbf{A}$ is the projection matrix for a subspace $\mathcal{U}$ of $\mathbb {R}^{n \times 1}$ along a subspace $\mathcal{V}$ of $\mathbb {R}^{n \times 1}$ (where $\mathcal{U} \oplus \mathcal{V} = \mathbb {R}^{n \times 1}$ ), then $\mathbf{A}'$ is the projection matrix for $\mathcal{V}^{\perp }$ along $\mathcal{U}^{\perp }$ [where $\mathcal{U}^{\perp }$ and $\mathcal{V}^{\perp }$ are the orthogonal complements (with respect to the usual inner product and relative to $\mathbb {R}^{n \times 1}$ ) of $\mathcal{U}$ and $\mathcal{V}$ , respectively].

?

Problem 17.16

Show that, for any $n \times p$ matrix $\mathbf{X}$ , $\mathbf{X}\mathbf{X}^{-}$ is the projection matrix for $C(\mathbf{X})$ along $\mathcal{N}(\mathbf{X}\mathbf{X}^{-})$ .

?

Problem 17.17

(a)(b)

Let $\mathbf{Y}$ represent a matrix in a linear space $\mathcal{V}$ of $m \times n$ matrices, and let $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ represent subspaces of $\mathcal{V}$ . Adopting the terminology and using the results of Exercise 6, show that if $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ are independent and if $\mathcal{U}_{1} + \cdots + \mathcal{U}_{k} = \mathcal{V}$ , then

?

(a)

there exist unique matrices $\mathbf{Z}_{1}, \ldots , \mathbf{Z}_{k}$ in $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ , respectively, such that $\mathbf{Y} = \mathbf{Z}_{1} + \cdots + \mathbf{Z}_{k}$ and

(b)

for $i = 1, \ldots , k$ , $\mathbf{Z}_{i}$ equals the projection of $\mathbf{Y}$ on $\mathcal{U}_{i}$ along $\mathcal{U}_{1} + \cdots + \mathcal{U}_{i-1} + \mathcal{U}_{i+1} + \cdots + \mathcal{U}_{k}$ .

Problem 17.18

(a)(b)

Let $\mathcal{U}$ and $\mathcal{W}$ represent essentially disjoint subspaces (of $\mathbb {R}^{n \times 1}$ ) whose sum is $\mathbb {R}^{n \times 1}$ , and let $\mathbf{U}$ represent any $n \times s$ matrix such that $C(\mathbf{U}) = \mathcal{U}$ and $\mathbf{W}$ any $n \times t$ matrix such that $C(\mathbf{W}) = \mathcal{W}$ .

?

(a)

Show that the $n \times (s+t)$ partitioned matrix $(\mathbf{U}, \mathbf{W})$ has a right inverse.

(b)

Taking $\mathbf{R}$ to be an arbitrary right inverse of $(\mathbf{U}, \mathbf{W})$ and partitioning $\mathbf{R}$ as $\mathbf{R} = \begin{pmatrix} \mathbf{R}_{1} \\ \mathbf{R}_{2} \end{pmatrix}$ (where $\mathbf{R}_{1}$ has $s$ rows), show that the projection matrix for $\mathcal{U}$ along $\mathcal{W}$ equals $\mathbf{U}\mathbf{R}_{1}$ and that the projection matrix for $\mathcal{W}$ along $\mathcal{U}$ equals $\mathbf{W}\mathbf{R}_{2}$ .

Problem 17.19

(a)(b)

Let $\mathbf{A}$ represent the ( $n \times n$ ) projection matrix for a subspace $\mathcal{U}$ of $\mathbb {R}^{n \times 1}$ along a subspace $\mathcal{V}$ of $\mathbb {R}^{n \times 1}$ (where $\mathcal{U} \oplus \mathcal{V} = \mathbb {R}^{n \times 1}$ ), let $\mathbf{B}$ represent the ( $n \times n$ ) projection matrix for a subspace $\mathcal{W}$ of $\mathbb {R}^{n \times 1}$ along a subspace $\mathcal{X}$ of $\mathbb {R}^{n \times 1}$ (where $\mathcal{W} \oplus \mathcal{X} = \mathbb {R}^{n \times 1}$ ), and suppose that $\mathbf{A}$ and $\mathbf{B}$ commute (i.e., that $\mathbf{B}\mathbf{A} = \mathbf{A}\mathbf{B}$ ).

?

(a)

Show that $\mathbf{A}\mathbf{B}$ is the projection matrix for $\mathcal{U} \cap \mathcal{W}$ along $\mathcal{V} + \mathcal{X}$ .

(b)

Show that $\mathbf{A} + \mathbf{B} - \mathbf{A}\mathbf{B}$ is the projection matrix for $\mathcal{U} + \mathcal{W}$ along $\mathcal{V} \cap \mathcal{X}$ .

Remark.

?

for Part (b). Observe that $\mathbf{I} - (\mathbf{A} + \mathbf{B} - \mathbf{A}\mathbf{B}) = (\mathbf{I} - \mathbf{A})(\mathbf{I} - \mathbf{B})$ , and make use of Part (a).

Problem 17.20

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

Let $\mathcal{V}$ represent a linear space of $n$ -dimensional column vectors, and let $\mathcal{U}$ and $\mathcal{W}$ represent essentially disjoint subspaces whose sum is $\mathcal{V}$ . Then, an $n \times n$ matrix $\mathbf{A}$ is said to be a projection matrix for $\mathcal{U}$ along $\mathcal{W}$ if $\mathbf{A}\mathbf{y}$ is the projection of $\mathbf{y}$ on $\mathcal{U}$ along $\mathcal{W}$ for every $\mathbf{y} \in \mathcal{V}$ -- this represents an extension of the definition of a projection matrix for $\mathcal{U}$ along $\mathcal{W}$ in the special case where $\mathcal{V} = \mathbb {R}^{n}$ . Further, let $\mathbf{U}$ represent an $n \times s$ matrix such that $C(\mathbf{U}) = \mathcal{U}$ , and let $\mathbf{W}$ represent an $n \times t$ matrix such that $C(\mathbf{W}) = \mathcal{W}$ .

?

(a)

Show that an $n \times n$ matrix $\mathbf{A}$ is a projection matrix for $\mathcal{U}$ along $\mathcal{W}$ if and only if $\mathbf{A}\mathbf{U} = \mathbf{U}$ and $\mathbf{A}\mathbf{W} = \mathbf{0}$ or, equivalently, if and only if $\mathbf{A}'$ is a solution to the linear system $\begin{pmatrix} \mathbf{U}' \\ \mathbf{W}' \end{pmatrix}\mathbf{B} = \begin{pmatrix} \mathbf{U}' \\ \mathbf{0} \end{pmatrix}$ (in an $n \times n$ matrix $\mathbf{B}$ ).

(b)

Establish the existence of a projection matrix for $\mathcal{U}$ along $\mathcal{W}$ .

(c)

Show that if $\mathbf{A}$ is a projection matrix for $\mathcal{U}$ along $\mathcal{W}$ , then $\mathbf{I} - \mathbf{A}$ is a projection matrix for $\mathcal{W}$ along $\mathcal{U}$ .

(d)

Let $\mathbf{X}$ represent any $n \times p$ matrix whose columns span $\mathcal{N}(\mathbf{W}')$ or, equivalently, $\mathcal{W}^{\perp }$ . Show that an $n \times n$ matrix $\mathbf{A}$ is a projection matrix for $\mathcal{U}$ along $\mathcal{W}$ if and only if $\mathbf{A}' = \mathbf{X}\mathbf{R}_{*}$ for some solution $\mathbf{R}_{*}$ to the linear system $\mathbf{U}'\mathbf{X}\mathbf{R} = \mathbf{U}'$ (in a $p \times n$ matrix $\mathbf{R}$ ).

Problem 17.21

Let $\mathcal{U}_{1}, \ldots , \mathcal{U}_{k}$ represent independent subspaces of $\mathbb {R}^{n \times 1}$ such that $\mathcal{U}_{1} + \cdots + \mathcal{U}_{k} = \mathbb {R}^{n \times 1}$ (where the independence of subspaces is as defined in Exercise 6). Further, letting $s_{i} = \dim (\mathcal{U}_{i})$ (and supposing that $s_{i} > 0$ ), take $\mathbf{U}_{i}$ to be any $n \times s_{i}$ matrix such that $C(\mathbf{U}_{i}) = \mathcal{U}_{i}$ ( $i = 1, \ldots , k$ ). And, define $\mathbf{B} = (\mathbf{U}_{1}, \ldots , \mathbf{U}_{k})^{-1}$ , partition $\mathbf{B}$ as $\mathbf{B} = \begin{pmatrix} \mathbf{B}_{1} \\ \vdots \\ \mathbf{B}_{k} \end{pmatrix}$ (where, for $i = 1, \ldots , k$ , $\mathbf{B}_{i}$ has $s_{i}$ rows), and let $\mathbf{H} = \mathbf{B}'\mathbf{B}$ or (more generally) let $\mathbf{H}$ represent any matrix of the form

\mathbf{H} = \mathbf{B}_{1}'\mathbf{A}_{1}\mathbf{B}_{1} + \mathbf{B}_{2}'\mathbf{A}_{2}\mathbf{B}_{2} + \cdots + \mathbf{B}_{k}'\mathbf{A}_{k}\mathbf{B}_{k}, \tag {E.3}

where $\mathbf{A}_{1}, \mathbf{A}_{2}, \ldots , \mathbf{A}_{k}$ are symmetric positive definite matrices.

?

(a)

Using the result of Part (g)-(1) of Exercise 6 (or otherwise), verify that the partitioned matrix $(\mathbf{U}_{1}, \ldots , \mathbf{U}_{k})$ is nonsingular (i.e., is square and of rank $n$ ).

(b)

Show that $\mathbf{H}$ is positive definite.

(c)

Show that (for $j \neq i = 1, \ldots , k$ ) $\mathcal{U}_{i}$ and $\mathcal{U}_{j}$ are orthogonal with respect to $\mathbf{H}$ .

(d)

Using the result of Part (a)-(2) of Exercise 3 (or otherwise), show that, for $i = 1, \ldots , k$ , (1) $\mathcal{U}_{1} + \cdots + \mathcal{U}_{i-1} + \mathcal{U}_{i+1} + \cdots + \mathcal{U}_{k}$ equals the orthogonal complement $\mathcal{U}_{i}^{\perp }$ of $\mathcal{U}_{i}$ (where the orthogonality in the orthogonal complement is with respect to the bilinear form $\mathbf{x}'\mathbf{H}\mathbf{y}$ ) and (2) the projection of any $n \times 1$ vector $\mathbf{y}$ on $\mathcal{U}_{i}$ along $\mathcal{U}_{1} + \cdots + \mathcal{U}_{i-1} + \mathcal{U}_{i+1} + \cdots + \mathcal{U}_{k}$ equals the orthogonal projection of $\mathbf{y}$ on $\mathcal{U}_{i}$ with respect to $\mathbf{H}$ .

(e)

Show that if, for $j \neq i = 1, \ldots , k$ , $\mathcal{U}_{i}$ and $\mathcal{U}_{j}$ are orthogonal with respect to some symmetric positive definite matrix $\mathbf{H}_{*}$ , then $\mathbf{H}_{*}$ is expressible in the form (E.3).