eigenvalue decomposition using matlab
By Jessica Wood •
I would like to diagnolize a rank-1 matrix using the well known eigenvalue decomposition as$\mathbf{U}^H\mathbf{A}\mathbf{U} = diag (M, 0,\cdots, 0)$, where $\mathbf{A}$ is a Hermitian matrix and $\mathbf{U}$ is a unitary matrix.
While if we use eig.m in matlab, the resulting $\mathbf{U}$ fails to satisfy the property of unitary matrix $\mathbf{U}^H\mathbf{U} = \mathbf{I}_{M}$ due to the unit round-off error and the use of \pi.
What should I do?
The code can be given:
clc clear all close all c = 340; j = sqrt(-1); theta_s = 0; theta_s = theta_s*pi/180; M = 6; delta = 0.012; M_vector = (0:M-1)'; theta = 0:0.1:360; theta = theta*pi/180; f = 4000; %% eigenvalue decomposition of matrix omega = 2*pi*f; d = exp(-j*omega*M_vector*delta/c); h_DS = 1/M*d; A = d*d'; [U,D] = eig(A);1 Answer
$\begingroup$Due to round-off errors, MATLAB isn't recognizing that your matrix is Hermitian. You can fix this easily by averaging A with its Hermitian transpose:
>> ishermitian(A)
ans = logical 0
>> B=(A+A')/2;
>> ishermitian(B)
ans = logical 1Once you've done this, U'*U=I as expected.
>> [U,D]=eig(B);
>> U'*U
ans = Columns 1 through 4 1.0000 + 0.0000i -0.0000 + 0.0000i 0.0000 + 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i 1.0000 + 0.0000i -0.0000 - 0.0000i -0.0000 - 0.0000i 0.0000 - 0.0000i -0.0000 + 0.0000i 1.0000 + 0.0000i 0.0000 - 0.0000i -0.0000 + 0.0000i -0.0000 + 0.0000i 0.0000 + 0.0000i 1.0000 + 0.0000i 0.0000 - 0.0000i 0.0000 - 0.0000i 0.0000 - 0.0000i -0.0000 + 0.0000i -0.0000 + 0.0000i -0.0000 - 0.0000i 0.0000 - 0.0000i 0.0000 + 0.0000i Columns 5 through 6 0.0000 + 0.0000i -0.0000 - 0.0000i 0.0000 + 0.0000i -0.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 + 0.0000i -0.0000 - 0.0000i 0.0000 - 0.0000i 1.0000 + 0.0000i 0.0000 + 0.0000i 0.0000 - 0.0000i 1.0000 + 0.0000i
