MATLAB：如何从特定的复杂向量中制作Hermitean矩阵？

Question

Is given :
stationary mass ms=1;
Eta-constant eta=0.45;
variable number of repetitions, eg N=5;
omega OM=sqrt(ks/ms);
angular frequency om=eta*OM;
time period T=2*pi/om;
upper bound TTT=1.5;
variable for creating function t=0:0.001:TTT;

I made a function like that:

kt=zeros(size(t));
for j=1:2*N+1
    n= j-(N+1);    
    if n==0
        k(j)=ks/2;
    else
        k(j)=i/pi/n;
    end
    kt=kt+k(j)*exp(i*n*om*t);
end

It's a Sawtooth wave and there is my problem . From the complex vector kt with value 1x1501 double I have to make the Hermitean matrix for variable N . This means that N can be 5, can be 50, 100, etc. The matrix should look like (picture):

Where k1 is k for N=1, k0 is k for N=0 or k-1 is k for N=-1. Size of matrix is 2*N+1 and 2*N+1.

Thank you for your help and responding!

Answer 1

That's a Toeplitz matrix , you can use the toeplitz command to generate the matrix above. In the general case, this would have been written as:

H = toeplitz(kt(N:end), kt(1:N + 1))

where the first N values in kt correspond to k _{- N} , ... k _-1 , and the last N + 1 values are k ₀ , ... k _N . However, since H is Hermitian, this can be simplified to:

H = toeplitz(kt(N:end));

Answer 2

Try this code:

k=[1 2+i 3+i 4+i 5+i];
N=7;
M=diag(k(1)*ones(N,1));

for j=1:length(k)-1
    M=M+diag(k(j+1)*ones(N-j,1),j)+diag(conj(k(j+1))*ones(N-j,1),-j)
end;

Here N should be equal or greater than the length of k array