Solving Eigenvalue Problems

Solving Eigenvalue Problems-71
Most commonly, the eigenvalue sequences are expressed as sequences of similar matrices which converge to a triangular or diagonal form, allowing the eigenvalues to be read easily.The eigenvector sequences are expressed as the corresponding similarity matrices.

Most commonly, the eigenvalue sequences are expressed as sequences of similar matrices which converge to a triangular or diagonal form, allowing the eigenvalues to be read easily.The eigenvector sequences are expressed as the corresponding similarity matrices.

Hessenberg and tridiagonal matrices are the starting points for many eigenvalue algorithms because the zero entries reduce the complexity of the problem.

Several methods are commonly used to convert a general matrix into a Hessenberg matrix with the same eigenvalues.

So the algebraic multiplicity is the multiplicity of the eigenvalue as a zero of the characteristic polynomial.

Since any eigenvector is also a generalized eigenvector, the geometric multiplicity is less than or equal to the algebraic multiplicity.

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.

Any problem of numeric calculation can be viewed as the evaluation of some function ƒ for some input of the problem is the ratio of the relative error in the function's output to the relative error in the input, and varies with both the function and the input.

The condition number describes how error grows during the calculation.

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.

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