Hessenberg matrix reduction

15.2.12 Hessenberg matrix reduction

A Hessenberg matrix is a square matrix where the coefficients below the sub-principal diagonal are all zeros. The hessenberg command finds a Hessenberg matrix equivalent to a given square matrix.

hessenberg takes one mandatory argument and one optional argument:
- A, a matrix.
- n, an integer, either 0, −1, −2 or a prime number greater than 1 (by default n=0).
hessenberg(A ⟨,n⟩) returns a list [P,B] with B=P⁻¹AP and:
- if n=0, B is a Hessenberg matrix.
- if n=−1, the calculations are approximate and B is upper triangular.
- if n=−2, the calculations are approximate, P is orthogonal and B has zero sub-subdiagonal elements.

SCHUR(A) is equivalent to hessenberg(A,-1), which is compatible with HP calculators.

Example

Let

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3	2	2	2	2
2	1	2	−1	−1
2	2	1	−1	1
2	−1	−1	3	1
2	−1	1	1	2

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Input:

A:=[[3,2,2,2,2],[2,1,2,-1,-1],[2,2,1,-1,1],[2,-1,-1,3,1],[2,-1,1,1,2]]; P,B:=hessenberg(A)

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−

−1

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Indeed, pcar(A) and pcar(B) both return [1,−10,13,71,−50,−113] and it is easily verified that B=P⁻¹AP.

B:=epsilon2zero(hessenberg(A,-1))

Output (to 2 digits):

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0.729361953258	−0.420012536934	−0.509269713814	−0.170245642208	−0.057283416961
0.24959754936	0.716215130873	−0.0478044979307	−0.376346706145	−0.529919650904
0.354791329292	−0.301345983892	0.742917332375	0.18980875771	−0.441995682565
0.340274973326	0.168149195581	0.427145129505	−0.462826562173	0.67770008281
0.405053403494	0.437654888177	−0.0630869571782	0.761014971552	0.247520076117

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6.70109038302	0	0	0	0
0	−1.86020364148	0	0	0
0	0	−1.15956801687	0	0
0	0	0	1.68836042443	0
0	0	0	0	4.6303208509

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