Eigenvalues corresponds to eigenvectors

In matlab, the command [V,L]=eig(h) produces the eigenvectors and eigenvalues of the square matirx h. But I would like to know in which order this eigenvectors appear? I mean how can I observe that which eigenvalues corresponds to which eigenvectors. I am really confused at this point. Pl somebody help me to understand this. Here I have taken an example.

clc;clear;
syms a b c
h=[a 0 1 0;0 b 2 0;1 0 c 0;0 3 0 a];
[V,L]=eig(h)

This produces the output as

 
V =
 
[ 0,         0,                                a/12 - b/6 + c/12 - (a^2 - 2*a*c + c^2 + 4)^(1/2)/12,                                a/12 - b/6 + c/12 + (a^2 - 2*a*c + c^2 + 4)^(1/2)/12]
[ 0, b/3 - a/3,                                         c/6 - a/6 - (a^2 - 2*a*c + c^2 + 4)^(1/2)/6,                                         c/6 - a/6 + (a^2 - 2*a*c + c^2 + 4)^(1/2)/6]
[ 0,         0, (a*b)/6 - (a*c)/6 - (b/6 - c/6)*(a/2 + c/2 - (a^2 - 2*a*c + c^2 + 4)^(1/2)/2) + 1/6, (a*b)/6 - (a*c)/6 - (b/6 - c/6)*(a/2 + c/2 + (a^2 - 2*a*c + c^2 + 4)^(1/2)/2) + 1/6]
[ 1,         1,                                                                                   1,                                                                                   1]
 
 
L =
 
[ a, 0,                                           0,                                           0]
[ 0, b,                                           0,                                           0]
[ 0, 0, a/2 + c/2 - (a^2 - 2*a*c + c^2 + 4)^(1/2)/2,                                           0]
[ 0, 0,                                           0, a/2 + c/2 + (a^2 - 2*a*c + c^2 + 4)^(1/2)/2]
 

But how do I associate the eigenvector with its corresponding eigenvealue.

Answer 1

[Vladimir Sovkov]

Absolutely standard: L(k,k) ~ V(:,k).

You can check it with the code:

for k=1:size(h,1)
disp(strcat('k=',num2str(k),'; h*v-lambda*v=',num2str(double(norm(simplify(h*V(:,k) - L(k,k)*V(:,k)))))));
end

[Vladimir Sovkov]

Notice that your matrix is not even symmetric, and a, b, c are considered arbitrary complex numbers. Some extra assumptions can simplify the result.

[AVM]

Thanks for your reply. Sorry,I didn't get your point. I just need to know for which eigenvalue, what is its corresponding eigenvector. I am runnig your code but what is tets it signifies. Is it that characteristics equation? Pl tell me in detail if possible.

[Vladimir Sovkov]

It is not the characteristic equation. It just tests the straightforward definition of the eigenvalues and eigenvectors: (v is the eigenvector and λ is the eigenvalue).

[AVM]

@Vladimir: Thanks for your clarification. Now, I got the point. It is to test that \nu is an eigenvector with corresponding eigenvalues \lambda. It's basically that eigenvalue equation.

[AVM]

@Vladimir: Is this eigenvector is normalized to unity? Or I have to make them normalize using V(:,k) / norm(V:,k)) ? Pl inform me.

[Vladimir Sovkov]

Generally, not normalized. I have noticed that the eigenvectors are normalized for real symmetric matrices but not in a general case.

[AVM]

Thanks for your reply.

[AVM]

@Vladimir: Is there any way to call c particular compoment of a column vector? I mean to say how can I call a component of column matrix in matlab? Here, I taken an example,

syms a b c d
h=[a;b;c;d]

How can I call 'c' element? What is the corresponding command? Pl help me.

[AVM]

And what about in this case also?

clc;
clear;
syms a b c 
assume(a,'real');
assume(b,'real');
assume(c,'real');
h=[a 1 0;3 b 1;1 0 c];
[V,L]=eig(h);
u=simplify(V(:,1),'Steps',50)

The u is here

 
u =
 
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                           a/3 + b/3 - (2*c)/3 + ((((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)^3 + ((a + b + c)^3/27 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 - (3*c)/2 + (a*b*c)/2 + 1/2)^2)^(1/2) - (3*c)/2 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 + (a + b + c)^3/27 + (a*b*c)/2 + 1/2)^(1/3) - ((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)/((((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)^3 + ((a + b + c)^3/27 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 - (3*c)/2 + (a*b*c)/2 + 1/2)^2)^(1/2) - (3*c)/2 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 + (a + b + c)^3/27 + (a*b*c)/2 + 1/2)^(1/3)
 a*c + (a/3 + b/3 + c/3 + ((((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)^3 + ((a + b + c)^3/27 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 - (3*c)/2 + (a*b*c)/2 + 1/2)^2)^(1/2) - (3*c)/2 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 + (a + b + c)^3/27 + (a*b*c)/2 + 1/2)^(1/3) - ((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)/((((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)^3 + ((a + b + c)^3/27 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 - (3*c)/2 + (a*b*c)/2 + 1/2)^2)^(1/2) - (3*c)/2 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 + (a + b + c)^3/27 + (a*b*c)/2 + 1/2)^(1/3))^2 - (a + c)*(a/3 + b/3 + c/3 + ((((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)^3 + ((a + b + c)^3/27 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 - (3*c)/2 + (a*b*c)/2 + 1/2)^2)^(1/2) - (3*c)/2 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 + (a + b + c)^3/27 + (a*b*c)/2 + 1/2)^(1/3) - ((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)/((((a*b)/3 + (a*c)/3 + (b*c)/3 - (a + b + c)^2/9 - 1)^3 + ((a + b + c)^3/27 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 - (3*c)/2 + (a*b*c)/2 + 1/2)^2)^(1/2) - (3*c)/2 - ((a + b + c)*(a*b + a*c + b*c - 3))/6 + (a + b + c)^3/27 + (a*b*c)/2 + 1/2)^(1/3))
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                     1

Now if I would like to call 2nd component of this column matrix u, then what should I have to do?

[Vladimir Sovkov]

The array indexing is described at https://www.mathworks.com/help/matlab/math/array-indexing.html?searchHighlight=matrix%20indexing&s_tid=doc_srchtitle

E.g., you address the entire k-th column of a matrix as V(:,k).