Q73E Let n be an even integer.In both... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 5: Q73E (page 236)

Let n be an even integer.In both parts of this problem,let Vbe the subspace of all vector $\overset{鈫�}{x}$ in $R^{n}$
such that $x_{j + 2} = x_{j} + x_{j + 1} 鈭赌 j = 1, 2, . ., n 鈭� 2 .$ .Consider the basis $\overset{鈫�}{v}, \overset{鈫�}{w}$ of V with
$\overset{鈫�}{a} = [\begin{array}{l} 1 \\ a \\ a^{2} \\ . \\ . \\ . \\ a^{n 鈭� 1} \end{array}], \overset{鈫�}{b} = [\begin{array}{l} 1 \\ b \\ b^{2} \\ . \\ . \\ . \\ b^{n 鈭� 1} \end{array}]$
where $a = \frac{1 + \sqrt{5}}{2}$ and $b = \frac{1 鈭� \sqrt{5}}{2}$
a.Show that $\overset{鈫�}{a} 鈥�$ is orthogonal to $\overset{鈫�}{b}$
b.Explain why the matrix P of the orthogonal projection onto V is a Hankel matrix.

Short Answer

Expert verified

a.The solution is orthogonal.

b.It suffices to show that M and N are Hankel matrices.Indeed $m_{i j} = a^{i + j 鈭� 2} = m_{i + j, j 鈭� 1}$ and $n_{i j} = b^{i + j 鈭� 2} = n_{i + 1, j 鈭� 1}$ for all $i = 1, n 鈭� 1 鈥夆赌夆赌夆赌夆赌� a n d 鈥夆赌� j = 2, 3, ..., n$

Step by step solution

(a)Step1:Definition for orthogonal

Two vectors $\overset{鈫�}{v}$ and $\overset{鈫�}{w}$ in $R^{n}$ are called orthogonal if $\overset{鈫�}{v} . \overset{鈫�}{w} = 0$

Verification of orthogonality

Assume n be an even integer.

let V be the subspace of all vector $\overset{鈫�}{x}$ in $R^{n}$ such that $x_{j + 2} = x_{j} + x_{j + 1} 鈭赌 j = 1, 2, .., n 鈭� 2.$ .Consider the basis $\overset{鈫�}{v}, \overset{鈫�}{w}$ of V with:

$\overset{鈫�}{a} = [\begin{array}{l} 1 \\ a \\ a^{2} \\ . \\ . \\ . \\ a^{n 鈭� 1} \end{array}], \overset{鈫�}{b} = [\begin{array}{l} 1 \\ b \\ b^{2} \\ . \\ . \\ . \\ b^{n 鈭� 1} \end{array}]$

Where $a = \frac{1 + \sqrt{5}}{2}$ and $b = \frac{1 鈭� \sqrt{5}}{2}$

Note that ab=-1

Now we have to prove that $\overset{鈫�}{a} . \overset{鈫�}{b} = 0$ for the verification of orthogonality.

$\begin{matrix} \overset{鈫�}{a} = {鈭�}_{k = 0}^{n 鈭� 1} a^{k} \\ \overset{鈫�}{b} = {鈭�}_{k = 0}^{n 鈭� 1} b^{k} \\ \overset{鈫�}{a} . \overset{鈫�}{b} = {鈭�}_{k = 0}^{n 鈭� 1} a^{k} {鈭�}_{k = 0}^{n 鈭� 1} b^{k} \\ \overset{鈫�}{a} . \overset{鈫�}{b} = {鈭�}_{k = 0}^{n 鈭� 1} a^{k} b^{k} \end{matrix}$

Further calculation can be as follows:

$\begin{matrix} \overset{鈫�}{a} . \overset{鈫�}{b} = {鈭�}_{k = 0}^{n 鈭� 1} {(a b)}^{k} \\ 鈥夆赌夆赌夆赌夆赌夆€夆赌夆€夆赌� = \frac{1 鈭� {(a b)}^{n}}{1 鈭� a b} \\ 鈥夆赌夆赌夆赌夆赌夆€夆赌夆€� = \frac{1 鈭� {(鈭� 1)}^{n}}{1 鈭� (鈭� 1)} \\ 鈥夆赌夆赌夆赌夆赌夆€夆赌� = 0 鈥夆赌夆赌夆赌夆赌夆€� 鈭赌鈥夆赌夆€夆赌� n \end{matrix}$

Thus $\overset{鈫�}{a} . \overset{鈫�}{b} = 0$ for all even integer n.

Hence $\overset{鈫�}{a}$ and $\overset{鈫�}{b}$ are orthogonal.

Therefore, the solution.

(b)Step 3:Explanation of orthogonal projection onto Hankel matrix

By a theorem we know that a subspace V of $R^{n}$ with orthonormal basis ${\overset{鈫�}{u}}_{1}, {\overset{鈫�}{u}}_{2}, ..., {\overset{鈫�}{u}}_{m}$ .The matrix P of the orthogonal projection onto V is P=QQ^T.

Where $Q = [\begin{array}{l} . 鈥夆赌夆赌夆赌夆赌夆€� . 鈥夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌� . \\ {\overset{鈫�}{u}}_{1} 鈥夆赌� {\overset{鈫�}{u}}_{2} 鈥夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌� {\overset{鈫�}{u}}_{m} \\ 鈥夆赌� . 鈥夆赌夆€夆赌� . 鈥夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆€夆赌夆赌夆赌夆赌夆€� . 鈥� \end{array}]$

Note that the matrix P is symmetric, since:

$\begin{matrix} P^{T} = {(Q Q^{T})}^{T} \\ 鈥夆赌夆赌夆赌夆赌夆€夆赌夆€� = Q^{T} {(Q^{T})}^{T} \\ 鈥夆赌夆赌夆赌夆赌夆€夆赌� = Q^{T} Q \\ 鈥夆赌夆赌夆赌夆赌夆€� = P \end{matrix}$

Thus, the preceding paragraph P is a linear combination of the matrices $M = \overset{鈫�}{a} {\overset{鈫�}{a}}^{T}$ and $N = \overset{鈫�}{b} {\overset{鈫�}{b}}^{T}$ .

It suffices to show that M and N are Hankel matrices.

Indeed $m_{i j} = a^{i + j 鈭� 2} = m_{i + j, j 鈭� 1}$ and $n_{i j} = b^{i + j 鈭� 2} = n_{i + 1, j 鈭� 1}$ for all $i = 1, n 鈭� 1 鈥夆赌夆赌夆赌夆赌� a n d 鈥夆赌� j = 2, 3, ..., n$

Hence the explanation.

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91影视

Short Answer

Step by step solution

(a)Step1:Definition for orthogonal

Verification of orthogonality

(b)Step 3:Explanation of orthogonal projection onto Hankel matrix

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