Mathematica 9 is now available
SEARCH MATHEMATICA 8 DOCUMENTATION
THIS IS DOCUMENTATION FOR AN OBSOLETE PRODUCT.
SEE THE DOCUMENTATION CENTER FOR THE LATEST INFORMATION.
Mathematica > Mathematics and Algorithms > Matrices and Linear Algebra > Matrix Decompositions >
Built-in Mathematica SymbolTutorials »|See Also »|More About »

HessenbergDecomposition

HessenbergDecomposition[m]
gives the Hessenberg decomposition of a matrix m.
  • The result is given in the form {p, h} where p is a unitary matrix such that p.h.Conjugate[Transpose[p]]Equalm.
  • The matrix m must be square.
EXAMPLESCLOSE ALL
Basic Examples   (1)
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:
In[1]:=
Click for copyable input
Out[1]=
The matrix h is an upper Hessenberg matrix:
In[2]:=
Click for copyable input
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:
Properties & Relations   (1)
A random 4×4 matrix:
Compute its Hessenberg decomposition:
The matrix p is unitary:
The matrix h is upper Hessenberg:
The original matrix m is given by p.h.ConjugateTranspose[p]:
Possible Issues   (1)
HessenbergDecomposition only works with matrices of approximate numerical values:
Use JordanDecomposition for exact matrices:
New in 5.1
Ask a question about this page  |  Suggest an improvement  |  Leave a message for the team