Using a variable in a matrix then plotting one element of the matrix.

Hello,

I'm trying to work with matrices and variables.

Ideally, what I would like to do would be to have a 4 by 4 Matrix (let's call it M for now).

Each of the elements of M depend on a variable x.

Now I would like to take the inverse of M and plot inv(M)(1,1).

When I try doing that in Matlab, I first define my variable

x=0:0.01:5;

Then calculate my matrix

M=A*x^2+B*x+C (where A, B, C are 4 by 4 matrices)

But when I do that, there's an error with matrix sizes.

Should I adapt the size of my M matrix, or am I doing this completely wrong?

Thanks in advance.

Answer 1

[Kye Taylor]

When matlab sees A.*B or A.^B the two arrays A and B need to be the same size. In the expression Z = -M.*Om.^2 + i.*Om.*C + K, the matrix M is 4-by-4 but Om is 1-by-501. The regular * and ^ also won't work in your situation because MATLAB interprets those operations as matrix multiplication and matrix exponentiation so the common rule from linear algebra of inner-dimensions is required. That is M*x only works for x that have dimensions 4-by-(someNumber).

Let me make an example that is similar to your situation. Let M be a 2-by-2 matrix whose entries depend on a parameter p. For example, M could have entries

M(1,1) = 1; M(1,2) = p; M(2,1) = p^2; M(2,2) = p; M(3,3) = 1; M(4,4) = 1; and zeros everywhere else.

I will define M in MATLAB using the command

M = @(q) [1,2*q,0,0;q^2,1,0,0;0,0,1,0;0,0,0,1];

In the MATLAB Workspace Window, notice that M is a function handle. But, if you do

M1 = M(1),

then M1 is a 4-by-4 double. (Search the documentation for function handle for more info.)

Now, I will plot the (1,1) element of M inverse as a function of the parameter p.

ps = 0:0.1:5;
oneOneElement = zeros(size(ps)); % allocate memory for the (1,1) element
for p = 1:length(ps)
  tempMatrix = inv(M(p));
  oneOneElement(p) = tempMatrix(1,1);
end
figure,plot(ps,oneOneElement,'k.-')

[Julien]

Thanks,

I tried it, and it seemed to work. (Not the expected result, but at least now I don't have an error anymore)

I didn't know about function handles...

I'm going to look more into details about it...

Thank you!

[Kye Taylor]

My pleasure. Function handles, of which M is known as an anonymous function, are super handy. Especially when you start getting fancy with arrayfun, bsxfun, etc. Check those out too.

Answer 2

[Walter Roberson]

M=A*x.^2+B*x+C

[Julien]

Hi,

Thanks, I had already tried that, and it didn't work. I also used .* but that didn't help...

My actual code is as follow:

Om = 0:0.01:5;

M = [1+mbar, mbar; 1, 1];

C = [2*ksi*w, 0; 0, 2*ksid*wd];

K = [w^2, 0; 0, wd^2];

Z = -M.*Om.^2 + i.*Om.*C + K;

H = inv(Z)

plot (Om,H(1,1))

[Walter Roberson]

Did you realize that the syntax 0:0.01:5 produces a row vector? So size 1 by something? When you have a 4 x 4 matrix matrix-multiplying a second matrix, then because inner dimensions need to match for matrix multiplication, the second matrix would have to be 4 x something. But yours is 1 by something instead.

Your Om is definitely not length 4.

Your Z = -M.*Om.^2 + i.*Om.*C + K; is being careless about whether Om is on the left-hand size of the multiplication or on the right hand side. It doesn't matter for .* of course, but is there any good reason not to be consistent in the placement of the coefficients, so that you would not have to re-arrange terms if you were to switch to * instead of .* ?

Okay, what I guess you want is to replace your current Z= line with

nOm = length(Om);

HOm = zeros(nOm,1);

for K = 1 : nOm

Z = -M .* Om(K).^2 + i .* C .* Om(k) + K;

H = inv(Z);

HOm = H(1,1);

end

plot(Om, HOm);