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Hessenberg Matrix


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)].

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.


See also

Hessenberg Decomposition, Toeplitz Matrix, Triangular Matrix

Portions 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 Matrix

Cite this as:

Dubrulle, Austin A. and Weisstein, Eric W. "Hessenberg Matrix." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HessenbergMatrix.html

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