A Hessenberg matrix is a matrix of the form
![[a_(11) a_(12) a_(13) ... a_(1(n-1)) a_(1n); a_(21) a_(22) a_(23) ... a_(2(n-1)) a_(2n); 0 a_(32) a_(33) ... a_(3(n-1)) a_(3n); 0 0 a_(43) ... a_(4(n-1)) a_(4n); 0 0 0 ... a_(5(n-1)) a_(5n); | | | | | |; 0 0 0 a_((n-1)(n-2)) a_((n-1)(n-1)) a_((n-1)n); 0 0 0 0 a_(n(n-1)) a_(nn)].](/images/equations/HessenbergMatrix/NumberedEquation1.svg) |
Hessenberg matrices were first investigated by Karl Hessenberg (1904-1959), a German engineer whose dissertation investigated the computation of eigenvalues and eigenvectors of linear operators.
Hessenberg Matrix
See also
Hessenberg Decomposition, Toeplitz Matrix, Triangular MatrixPortions of this entry contributed by Austin A. Dubrulle
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References
Hessenberg, K. Thesis. Darmstadt, Germany: Technische Hochschule, 1942.Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Reduction of a General Matrix to Hessenberg Form." §11.5 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 476-480, 1992.Referenced on Wolfram|Alpha
Hessenberg MatrixCite this as:
Dubrulle, Austin A. and Weisstein, Eric W. "Hessenberg Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HessenbergMatrix.html