Hessenberg
Performs a Hessenberg decomposition of a matrix.
Syntax
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[H, Q ≔ ]Hessenberg(A)-
Ais a matrix
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Description
If A is a matrix, then Hessenberg(A) performs a Hessenberg decomposition of A, that is, it finds a similar Hessenberg matrix H and a unitary matrix Q such that A = Q⋅H⋅Q*.
The result of Hessenberg(A) is an assignment list of length 2, containing H and Q.
Examples
A ≔ ❨❨6, 2, 4, 5, 1❩, ❨2, 8, −3, 1, 2❩, ❨6, 3, 3, −1, 1❩, ❨7, 5, −2, 6, 9❩, ❨4, 5, 2, 1, 3❩❩
⎛ 6 2 4 5 1⎞ ⎜ 2 8 −3 1 2⎟ ⎜ 6 3 3 −1 1⎟ ⎜ 7 5 −2 6 9⎟ ⎝ 4 5 2 1 3⎠
(H, Q) ≔ Hessenberg(A)
⎛ 6 −6.53853048876 −0.289489513914 1.73510386979 −0.391445309069⎞ ⎜ −10.246950766 8.07619047619 −3.10726843673 −0.135912197927 −9.30010937136⎟ ⎜−3.76620873764⋅10^−19 −3.40642317114 5.63166766609 −1.43813721491 2.0718580141⎟ ⎜−2.13956670128⋅10^−19 7.08790648724⋅10^−20 4.4791285155 0.924051521064 2.76729575219⎟ ⎝−4.23935425962⋅10^−19 1.20379404785⋅10^−19 −1.62630325873⋅10^−19 3.81293940331 5.36809033666⎠ ⎛ 1 0 0 0 0⎞ ⎜ 0 −0.19518001459 −0.090311997272 0.579441392757 0.786127329008⎟ ⎜ 0 −0.585540043769 −0.78661476778 −0.0734174438009 −0.181631338703⎟ ⎜ 0 −0.683130051064 0.557697046598 0.322918581587 −0.343556294099⎟ ⎝ 0 −0.390360029179 0.249108318759 −0.744702048454 0.480606858225⎠
IsUpperHessenberg(H) ∧ IsOrthogonal(Q)
true
Q⋅H⋅Q*
⎛ 6 2 4 5 1⎞ ⎜ 2 8 −3 1 2⎟ ⎜ 6 3 3 −1 1⎟ ⎜ 7 5 −2 6 9⎟ ⎝ 4 5 2 1 3⎠
See also
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Matrix decompositions (list)