• :00Days
  • :00Hours
  • :00Mins
  • 00Seconds
A new era for learning is coming soonSign up for free
Log In Start studying!

Select your language

Suggested languages for you:
Answers without the blur. Sign up and see all textbooks for free! Illustration


Linear Algebra With Applications
Found in: Page 393
Linear Algebra With Applications

Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974

Answers without the blur.

Just sign up for free and you're in.


Short Answer

Diagonalize the n×n matrix

[1000101010001000101010001] .

(All ones along both diagonals, and zeros elsewhere.)

If is even the Eigen basis is


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

See the step by step solution

Step by Step Solution

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×n matrix as:


We can write A as A =Jn+In , where

localid="1660201143401" Jn=[0000100010000000101010001]


Now consider the transpose of the matrix as represented below:


We can see that the matrix Jn is symmetric since Jn=JnT .

Now evaluate JnJnT as represented below:



The matrix Jn is orthogonal since JnJnT=JnJnT=In.

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 n2 and if is odd, then the eigenvalue 1 will be of multiplicity n+12 and eigen-value -1 will be of multiplicity n-12 .

The Eigen value of Jn will be +1 or -1 and when added with Eigen value of In which is +1, therefore the eigen values are 0 and 2 with multiplicities n2 .

The Eigen basis for the Eigen value is:

role="math" localid="1660204574030" e1+en+1; i=1,...,n2 And for the Eigen value 0 is :

e1-en+1; i=1,...,n2

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

e1-en,e2-en-1,...,en/2+1,e1+en,+e2+en-1,...,en/2+en/2+1 With associated eigenvalues 0( n/2 times) and 2(n/ 2times) .

Therefore, is even the Eigen basis is


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

Recommended explanations on Math Textbooks

94% of StudySmarter users get better grades.

Sign up for free
94% of StudySmarter users get better grades.