6 Consider the matrix A given by 10032 2 1 00 3 A=132100 0 321 0 003 2 1 (a) Use

Question

6 Consider the matrix A given by 10032 2 1 00 3 A=132100 0 321 0 003 2 1 (a) Use a numerical software package to find the eigendecomposition. A = EAE, where is the diagonal matrix of eigenvalues and the columns of E are the eigenvectors. (Suggestion, see eig in Matlab.) (i) Look at the scaled magnitudes of the eigenvector entries: abs (E) sqrt (5). What do you observe? (i) Look at the angles of the eigenvector entries: 180/pi*angle(E). What do you observe? (iii) Compare the conjugate transpose of E to the inverse of E. What do you observe? (b) What is the definition of "circulant" matrix? c)(for graduate students) What property must a circulant matrix satisfy in order that its eigenvalues are real-valued?

Explanation / Answer

Using MATLAB functions to solve the given solution,

A = [1 0 0 3 2;2 1 0 0 3;3 2 1 0 0;0 3 2 1 0;0 0 3 2 1];

(a)

[E,D] = eig(A)

Eigen vector matrix E =

[ 0.4472 + 0.0000i 0.4472 + 0.0000i 0.4472 + 0.0000i -0.3618 - 0.2629i -0.3618 + 0.2629i;

0.4472 + 0.0000i 0.1382 - 0.4253i 0.1382 + 0.4253i 0.4472 + 0.0000i 0.4472 + 0.0000i;

0.4472 + 0.0000i -0.3618 - 0.2629i -0.3618 + 0.2629i -0.3618 + 0.2629i -0.3618 - 0.2629i;

0.4472 + 0.0000i -0.3618 + 0.2629i -0.3618 - 0.2629i 0.1382 - 0.4253i 0.1382 + 0.4253i;

0.4472 + 0.0000i 0.1382 + 0.4253i 0.1382 - 0.4253i 0.1382 + 0.4253i 0.1382 - 0.4253i ]

Diagonal matrix D with eigen values on the diagonal D =

[

6.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i ;

0.0000 + 0.0000i -0.8090 + 3.6655i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i;

0.0000 + 0.0000i 0.0000 + 0.0000i -0.8090 - 3.6655i 0.0000 + 0.0000i 0.0000 +0.0000i;

0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.3090 + 1.6776i 0.0000 + 0.0000i;

0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i 0.3090 - 1.6776i; ]

(i)

A1 = abs(E)*sqrt(5)

[

1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000

1.0000 1.0000 1.0000 1.0000 1.0000 ]

(ii)

A2 = 180/pi * angle(E)

[

0 0 0 -144.0000 144.0000

0 -72.0000 72.0000 0 0

0 -144.0000 144.0000 144.0000 -144.0000

0 144.0000 -144.0000 -72.0000 72.0000

0 72.0000 -72.0000 72.0000 -72.0000 ]

(iii)

complex conjugate transpose of E

A3 = E'

[

0.4472 + 0.0000i 0.4472 + 0.0000i 0.4472 + 0.0000i 0.4472 + 0.0000i 0.4472 + 0.0000i

0.4472 + 0.0000i 0.1382 + 0.4253i -0.3618 + 0.2629i -0.3618 - 0.2629i 0.1382 - 0.4253i

0.4472 + 0.0000i 0.1382 - 0.4253i -0.3618 - 0.2629i -0.3618 + 0.2629i 0.1382 + 0.4253i

-0.3618 + 0.2629i 0.4472 + 0.0000i -0.3618 - 0.2629i 0.1382 + 0.4253i 0.1382 - 0.4253i

-0.3618 - 0.2629i 0.4472 + 0.0000i -0.3618 + 0.2629i 0.1382 - 0.4253i 0.1382 + 0.4253i

]

A4 = inv(E)

[

0.4472 - 0.0000i 0.4472 - 0.0000i 0.4472 - 0.0000i 0.4472 + 0.0000i 0.4472 + 0.0000i

0.4472 - 0.0000i 0.1382 + 0.4253i -0.3618 + 0.2629i -0.3618 - 0.2629i 0.1382 - 0.4253i

0.4472 + 0.0000i 0.1382 - 0.4253i -0.3618 - 0.2629i -0.3618 + 0.2629i 0.1382 + 0.4253i

-0.3618 + 0.2629i 0.4472 + 0.0000i -0.3618 - 0.2629i 0.1382 + 0.4253i 0.1382 - 0.4253i

-0.3618 - 0.2629i 0.4472 - 0.0000i -0.3618 + 0.2629i 0.1382 - 0.4253i 0.1382 + 0.425 ]

So complex conjugate and inverse of eigen vector matrix are same.

(b) A circulant matrix is a square matrix where each row is a single shifted version of the precedent row.

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