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hessfact

..  hessfact(A)

Compute the Hessenberg decomposition of ``A`` and return a ``Hessenberg`` object. If ``F`` is the factorization object, the unitary matrix can be accessed with ``F[:Q]`` and the Hessenberg matrix with ``F[:H]``. When ``Q`` is extracted, the resulting type is the ``HessenbergQ`` object, and may be converted to a regular matrix with :func:`full`.

Examples

The hessfact function in Julia computes the Hessenberg decomposition of a given matrix A and returns a Hessenberg object. The resulting object can be used to access the unitary matrix and the Hessenberg matrix.

julia> A = [2.0 1.0 0.0 0.0; 1.0 2.0 1.0 0.0; 0.0 1.0 2.0 1.0; 0.0 0.0 1.0 2.0];

julia> F = hessfact(A)
Base.LinAlg.Hessenberg{Float64,Array{Float64,2}}
Q factor:
4×4 LinearAlgebra.QRPackedQ{Float64,Array{Float64,2}}:
 -0.447214  -0.774597  0.447214  0.0     
 -0.447214   0.258199  0.774597  0.0     
 -0.447214   0.516398 -0.258199 -0.707107
 -0.447214   0.258199 -0.258199  0.707107
Hessenberg factor:
4×4 Array{Float64,2}:
 2.23607  2.23607  0.894427  0.447214
 2.23607  1.51657  0.894427  0.447214
 0.0      1.51657  0.894427  0.447214
 0.0      0.0      1.78885   1.78885

In the example above, the Hessenberg decomposition of matrix A is computed using hessfact. The resulting Hessenberg object F allows access to the unitary matrix (F[:Q]) and the Hessenberg matrix (F[:H]).

To convert the HessenbergQ object to a regular matrix, you can use the full function:

julia> Q = full(F[:Q])
4×4 Array{Float64,2}:
 -0.447214  -0.774597   0.447214  0.0     
 -0.447214   0.258199   0.774597  0.0     
 -0.447214   0.516398  -0.258199 -0.707107
 -0.447214   0.258199  -0.258199  0.707107

The resulting Q matrix is a regular matrix representation of the unitary matrix extracted from the Hessenberg object.

Note: The hessfact function is part of the LinearAlgebra module in Julia.

See Also

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