hessenberg: Hessenberg Matrix

In pracma: Practical Numerical Math Functions

Hessenberg Matrix

Description

Generates the Hessenberg matrix for A.

Usage

hessenberg(A)

Arguments

A	square matrix

Details

An (upper) Hessenberg matrix has zero entries below the first subdiagonal.

The function generates a Hessenberg matrix H and a unitary matrix P (a similarity transformation) such that A = P * H * t(P).

The Hessenberg matrix has the same eigenvalues. If A is symmetric, its Hessenberg form will be a tridiagonal matrix.

Value

Returns a list with two elements,

H	the upper Hessenberg Form of matrix A.
H	a unitary matrix.

References

Press, Teukolsky, Vetterling, and Flannery (2007). Numerical Recipes: The Art of Scientific Computing. 3rd Edition, Cambridge University Press. (Section 11.6.2)

See Also

householder

Examples

A <- matrix(c(-149,   -50,  -154,
               537,   180,   546,
               -27,    -9,   -25), nrow = 3, byrow = TRUE)
hb  <- hessenberg(A)
hb
## $H
##           [,1]         [,2]        [,3]
## [1,] -149.0000  42.20367124 -156.316506
## [2,] -537.6783 152.55114875 -554.927153
## [3,]    0.0000   0.07284727    2.448851
## 
## $P
##      [,1]       [,2]      [,3]
## [1,]    1  0.0000000 0.0000000
## [2,]    0 -0.9987384 0.0502159
## [3,]    0  0.0502159 0.9987384

hb$P %*% hb$H %*% t(hb$P)
##      [,1] [,2] [,3]
## [1,] -149  -50 -154
## [2,]  537  180  546
## [3,]  -27   -9  -25

pracma documentation built on Nov. 10, 2023, 1:14 a.m.