why not books
Golub, Gene H., and Charles F. Van Loan. Matrix computations. Vol. 3. JHU Press, 2012.
and if you need something more basic like understanding the difference between the Holder p-norm of a matrix and the Frobenious norm.
Stewart, Gilbert W. Introduction to matrix computations. Elsevier, 1973.
Although there are probably some exceptions, the symmetric square matrix is the most common in application in signal processing.
Typically, you calculate the eigenvectors and the eigenvalues follow. I'm not aware of any algorithms that just calculate the eigenvalues.
The eigenvalues are sometimes called the matrix spectrum.
It would actaully be hard to not find a numerical linear algebra library that didn't compute eigenvector and SVD of a square matrix.
A popular book for those who don't want to wade in deeply is
Press, William H., Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery. Numerical recipes 3rd edition: The art of scientific computing. Cambridge university press, 2007.