BUILT-IN MATHEMATICA SYMBOL

HessenbergDecomposition

HessenbergDecomposition[m]
gives the Hessenberg decomposition of a matrix m.

The result is given in the form where p is a unitary matrix such that p.h.Conjugate[Transpose[p]]==m.

Find the Hessenberg decomposition of a 4×4 matrix:

The matrix h is an upper Hessenberg matrix:

Find the Hessenberg decomposition of a 4×4 matrix:

Out[1]=

The matrix h is an upper Hessenberg matrix:

Out[2]//MatrixForm=

Hessenberg decomposition works for complex matrices:

A matrix with entries having 24-digit precision:

The Hessenberg decomposition is computed using 24 digit precision:

A 4×4 random symmetric matrix:

Compute its Hessenberg decomposition:

Do 100 iterations of the unshifted QR algorithm:

The eigenvalues lie along the diagonal:

A random 4×4 matrix:

Compute its Hessenberg decomposition:

The matrix

is unitary:

The matrix

is upper Hessenberg:

The original matrix

is given by p.h.ConjugateTranspose[p]:

HessenbergDecomposition only works with matrices of approximate numerical values:

Use JordanDecomposition for exact matrices: