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based on an earlier version of the Wolfram Language.
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# 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.
• The matrix m must be square.
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=
 Scope   (2)
Hessenberg decomposition works for complex matrices:
A matrix with entries having 24-digit precision:
The Hessenberg decomposition is computed using 24 digit precision:
 Applications   (1)
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:
New in 5.1