Solving Eigenvalue Problems

Solving Eigenvalue Problems-60
So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial.

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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.

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.

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.

This process can be repeated until all eigenvalues are found.

If an eigenvalue algorithm does not produce eigenvectors, a common practice is to use an inverse iteration based algorithm with Because the eigenvalues of a triangular matrix are its diagonal elements, for general matrices there is no finite method like gaussian elimination to convert a matrix to triangular form while preserving eigenvalues.

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.

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