MATLAB: How to find a linear parameter within a matrix by khowing some of its eigenvalues

eigen values

Note the order of Matrix is 1000*1000.

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 My matrix have the form of
syms s
A=rand(1000);
B=s*rand(1000);
C=rand(1000);
M=[A,B;B,C];
 I think the problem is clear.
I posed this question in maple forum they proposed a method but my machine could not solve it because of high order matrix. I guess, Matlab is powerful in numerical computations, so switched to Matlab. look at;
http://www.mapleprimes.com/questions/221584-How-To-Find-A-Linear-Parameter-Within

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MATLAB: I was trying to solve an eigenvalue problem of two Identical matrices computed in a different way? Surprisingly one solution is six to ten times faster than other (though the solutions are identical)? Can any body explain the reason

Eig is probably faster when the matrix is symmetric.

A = rand(1000); M = A'*(eye(1000))*A*1.1;
norm(M - M.')
ans =
     0
M = A'*(eye(1000)*1.1)*A;
norm(M - M.')
ans =
   1.5208e-12

So, lets prove that by forcing this matrix to be TRULY symmetric.

tic,eig(M);toc
Elapsed time is 0.807391 seconds.
Q = (M + M.')/2;
tic,eig(Q);toc
Elapsed time is 0.115139 seconds.

Yes, I know that mathematically, M is symmetric, and should be so in both cases. But you need to learn that floating point arithmetic, as computed on a modern CPU, using tools like the BLAS, is NOT truly mathematics. Things that may be true in mathematics are not always so.

The parser is probably smart enough to force a product like the first case to be perfectly symmetric, via clever use of the BLAS. In the second case, it was apparently not that smart, so it simply does the matrix multiplies. And when you fool with the order of operations in floating point arithmetic, there is no assurance that two results will be identical.

Welcome to the wonderful, wacky world of floating point arithmetic.

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