Book edition 5th

Author(s) Otto Bretscher

Pages 442 pages

ISBN 9780321796974

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Short Answer

Diagonalize the $n \times n$ matrix
$[\begin{matrix} 1 & 0 & 0 & \dots & 0 & 1 \\ 0 & 1 & 0 & \dots & 1 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 1 & 0 & \dots & 1 & 0 \\ 1 & 0 & 0 & \dots & 0 & 1 \end{matrix}]$ .
(All ones along both diagonals, and zeros elsewhere.)

If is even the Eigen basis is

${\vec{e}}_{1} - {\vec{e}}_{n}, {\vec{e}}_{2} - {\vec{e}}_{n - 1}, . . ., {\vec{e}}_{(n / 2) + 1,} {\vec{e}}_{1} + {\vec{e}}_{n}, + {\vec{e}}_{2} + {\vec{e}}_{n - 1}, . . ., {\vec{e}}_{n / 2} + {\vec{e}}_{(n / 2) + 1}$

With associated eigenvalues 0(n / 2 times ) and 2(n / 2times ).

See the step by step solution

Step by Step Solution

TABLE OF CONTENTS :

TABLE OF CONTENTS

Step 1: The matrix

A matrix is a rectangular array or table of numbers, symbols, or expressions that are organised in rows and columns to represent a mathematical object or an attribute of such an item in mathematics.
For example, is a two-row, three-column matrix

Step 2: Determine the Diagonalize matrix

Consider the $n \times n$ matrix as:

$A = [\begin{matrix} 1 & 0 & 0 & \dots & 0 & 1 \\ 0 & 1 & 0 & \dots & 1 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 1 & 0 & \dots & 1 & 0 \\ 1 & 0 & 0 & \dots & 0 & 1 \end{matrix}]$

We can write A as A = $J_{n} + I_{n}$ , where

localid="1660201143401" $J_{n} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 1 & 0 & \dots & 1 & 0 \\ 1 & 0 & 0 & \dots & 0 & 1 \end{matrix}]$

$I_{n} = [\begin{matrix} 1 & 0 & 0 & \dots & 0 & 0 \\ 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 1 \end{matrix}]$

Now consider the transpose of the matrix as represented below:

$J_{n}^{T} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 1 & 0 & \dots & 0 & 0 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{matrix}] \Rightarrow J_{n}^{T} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 1 & 0 & \dots & 0 & 0 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{matrix}] = J$

We can see that the matrix $J_{n}$ is symmetric since $J_{n} = J_{n}^{T}$ .

Now evaluate $J_{n} J_{n}^{T}$ as represented below:

$J_{n} J_{n}^{T} = [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 1 & 0 & \dots & 0 & 0 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{matrix}] [\begin{matrix} 0 & 0 & 0 & \dots & 0 & 1 \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 1 & 0 & \dots & 0 & 0 \\ 1 & 0 & 0 & \dots & 0 & 0 \end{matrix}]$

$J_{n} J_{n}^{T} = [\begin{matrix} 1 & 0 & 0 & \dots & 0 & 0 \\ 0 & 1 & 0 & \dots & 0 & 0 \\ 0 & 0 & 1 & \dots & 0 & 0 \\ ⋮ & ⋮ & ⋮ & ⋱ & ⋮ & ⋮ \\ 0 & 0 & 0 & \dots & 1 & 0 \\ 0 & 0 & 0 & \dots & 0 & 1 \end{matrix}] = I_{n}$

The matrix $J_{n}$ is orthogonal since $J_{n} J_{n}^{T} = J_{n} J_{n}^{T} = I_{n}$ .

In general, if a matrix is symmetric, it is orthogonally diagonalizable and all its eigenvalues are real.

If it is also orthogonal, its eigenvalues must be 1 or -1 .

For a matrix, if n is even, then both eigen-values will be of multiplicity $\frac{n}{2}$ and if is odd, then the eigenvalue 1 will be of multiplicity $\frac{n + 1}{2}$ and eigen-value -1 will be of multiplicity $\frac{n - 1}{2}$ .

The Eigen value of $J_{n}$ will be +1 or -1 and when added with Eigen value of $I_{n}$ which is +1, therefore the eigen values are 0 and 2 with multiplicities $\frac{n}{2}$ .

The Eigen basis for the Eigen value is:

role="math" localid="1660204574030" ${\vec{e}}_{1} + {\vec{e}}_{n + 1}; i = 1, . . ., \frac{n}{2}$ And for the Eigen value 0 is :

${\vec{e}}_{1} - {\vec{e}}_{n + 1}; i = 1, . . ., \frac{n}{2}$

If n is even the Eigen basis is as represented below:

Therefore, is even the Eigen basis is

With associated eigenvalues 0 (n/ 2times) and 2(n / 2times) .

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Diagonalize the n×n matrix [100⋯01010⋯10001⋯00⋮⋮⋮⋱⋮⋮010⋯10100⋯01] . (All ones along both diagonals, and zeros elsewhere.)

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