Matlab in Math 461, part five Eigenvalues and Eigenvectors, etc. If A is a square n × n matrix, then the command E = eig(A) will produce a vector E whose entries are the n eigenvalues of A (including multiplicities). Some eigenvalues may be complex and the real and imaginary parts will be given. To find the eigenvectors, use the command
The theory is to find values of x that make the matrix determinant equal to zero. But, as I said, the matrix is singular and det (A) is always zero in matlab. I also tried eig (A), but it gives me answers that, in addition to being very long and undisplayable, are equations that still contain the variable!
Eigenvalues of a large Matrix. Learn more about eigenvalue
To solve for the eigenvalues, we write the equation in the form (A- I)x =0 The equation shows that the eigenvectors x lie in the nullspace of A- I. From the theory of linear algebra, non-trivial solutions require that we choose so that matrix A- I hasa nullspace, that is, it must have determinant zero: det(A- I)=0.
Or another way to think about it is it's not invertible, or it has a determinant of 0. So lambda is the eigenvalue of A, if and only if, each of these steps are true. And this is true if and only if-- for some at non-zero vector, if and only if, the determinant of lambda times the identity matrix minus A is equal to 0. And that was our takeaway.