# Solving Eigenvalue Problems

But it is possible to reach something close to triangular.An upper Hessenberg matrix is a square matrix for which all entries below the subdiagonal are zero.

Some algorithms also produce sequences of vectors that converge to the eigenvectors.For this reason algorithms that exactly calculate eigenvalues in a finite number of steps only exist for a few special classes of matrices.For general matrices, algorithms are iterative, producing better approximate solutions with each iteration.While there is no simple algorithm to directly calculate eigenvalues for general matrices, there are numerous special classes of matrices where eigenvalues can be directly calculated.These include: Since the determinant of a triangular matrix is the product of its diagonal entries, if T is triangular, then , respectively.Therefore, a general algorithm for finding eigenvalues could also be used to find the roots of polynomials.The Abel–Ruffini theorem shows that any such algorithm for dimensions greater than 4 must either be infinite, or involve functions of greater complexity than elementary arithmetic operations and fractional powers.Its base-10 logarithm tells how many fewer digits of accuracy exist in the result than existed in the input. It reflects the instability built into the problem, regardless of how it is solved.No algorithm can ever produce more accurate results than indicated by the condition number, except by chance.The algebraic multiplicities sum up to It is possible for a real or complex matrix to have all real eigenvalues without being hermitian.For example, a real triangular matrix has its eigenvalues along its diagonal, but in general is not symmetric.