Math Problems Vault

Problem 6.1

Use the Schwarz inequality to show that, for any two matrices $A$ and $B$ in a linear space $\mathcal{V}$ ,

\left\| A+B\right\| \leq \left\| A\right\| + \left\| B\right\| ,

with equality holding if and only if $B = 0$ or $A = kB$ for some nonnegative scalar $k$ . (This inequality is known as the triangle inequality.)

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Problem 6.2

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

Letting $A$ , $B$ , and $C$ represent arbitrary matrices in a linear space $\mathcal{V}$ , show that

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(a)

$\delta (B,A) = \delta (A,B)$ , that is the distance between $B$ and $A$ is the same as that between $A$ and $B$ ;

(b)

\delta (A,B) > 0, \quad \text{if } A \neq B, \qquad = 0, \quad \text{if } A = B,

that is, the distance between any two matrices is greater than zero, unless the two matrices are identical, in which case the distance between them is zero;

(c)

$\delta (A,B) \leq \delta (A,C) + \delta (C,B)$ , that is, the distance between $A$ and $B$ is less than or equal to the sum of the distances between $A$ and $C$ and between $C$ and $B$ ;

(d)

$\delta (A,B) = \delta (A+C, B+C)$ , that is, distance is unaffected by a translation of ``axes.''

Remark.

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For Part (c), use the result of Exercise 6.1, i.e., the triangle inequality.

Problem 6.3

(a)(b)

Let $\overrightarrow {w}_1'$ , $\overrightarrow {w}_2'$ , and $\overrightarrow {w}_3'$ represent the three linearly independent 4-dimensional row vectors $(6,0,-2,3)$ , $(-2,4,4,2)$ , and $(0,5,-1,2)$ , respectively, in the linear space $\mathbb {R}^{4}$ , and adopt the usual definition of inner product.

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(a)

Use Gram-Schmidt orthogonalization to find an orthonormal basis for the linear space $\mathrm{sp}(\overrightarrow {w}_1', \overrightarrow {w}_2', \overrightarrow {w}_3')$ .

(b)

Find an orthonormal basis for $\mathbb {R}^{4}$ that includes the three orthonormal vectors from Part (a). Do so by extending the results of the Gram-Schmidt orthogonalization [from Part (a)] to a fourth linearly independent row vector such as $(0,1,0,0)$ .

Problem 6.4

(a)(b)

Let $\left\{ A_1, \ldots , A_k\right\}$ represent a nonempty (possibly linearly dependent) set of matrices in a linear space $\mathcal{V}$ .

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(a)

Generalize the results underlying Gram-Schmidt orthogonalization (which are for the special case where the set $\left\{ A_1, \ldots , A_k\right\}$ is linearly independent) by showing (1) that there exist scalars $x_{ij}$ ( $i < j = 1, \ldots , k$ ) such that the set comprising the $k$ matrices

\begin{align} B_1 & = A_1, \\ B_2 & = A_2 - x_{12}B_1, \\ & \; \vdots \\ B_j & = A_j - x_{j-1,j}B_{j-1} - \cdots - x_{1j}B_1, \\ & \; \vdots \\ B_k & = A_k - x_{k-1,k}B_{k-1} - \cdots - x_{1k}B_1 \end{align}

is orthogonal; (2) that, for $j = 1, \ldots , k$ and for those $i < j$ such that $B_i$ is nonnull, $x_{ij}$ is given uniquely by

x_{ij} = \frac{A_j \bullet B_i}{B_i \bullet B_i};

and (3) that the number of nonnull matrices among $B_1, \ldots , B_k$ equals $\dim [\mathrm{sp}(A_1, \ldots , A_k)]$ .

(b)

Describe a procedure for constructing an orthonormal basis for $\mathrm{sp}(A_1, \ldots , A_k)$ .

Problem 6.5

Let $A$ represent an $m \times k$ matrix of rank $r$ (where $r$ is possibly less than $k$ ). Generalize the so-called QR decomposition of $A$ , which is for the special case where $r = k$ and is obtainable through the application of Gram-Schmidt orthogonalization to the columns of $A$ . Do so by using the results of Exercise 6.4 to obtain a decomposition of the form $A = QR_1$ , where $Q$ is an $m \times r$ matrix with orthonormal columns and $R_1$ is an $r \times k$ submatrix whose rows are the $r$ nonnull rows of a $k \times k$ upper triangular matrix $R$ having $r$ positive diagonal elements and $k-r$ null rows.

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