Hessenberg Matrix
Generates the Hessenberg matrix for A.
hessenberg(A)
A |
square matrix |
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.
Returns a list with two elements,
H |
the upper Hessenberg Form of matrix A. |
H |
a unitary matrix. |
Press, Teukolsky, Vetterling, and Flannery (2007). Numerical Recipes: The Art of Scientific Computing. 3rd Edition, Cambridge University Press. (Section 11.6.2)
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 -25Please choose more modern alternatives, such as Google Chrome or Mozilla Firefox.