Inlinear algebra, aHessenberg matrix is a special kind ofsquare matrix, one that is "almost"triangular. To be exact, anupper Hessenberg matrix has zero entries below the firstsubdiagonal, and alower Hessenberg matrix has zero entries above the firstsuperdiagonal.^[1] They are named afterKarl Hessenberg.^[2]

AHessenberg decomposition is amatrix decomposition of a matrix${\displaystyle A}$ into aunitary matrix${\displaystyle P}$ and a Hessenberg matrix${\displaystyle H}$ such that$${\displaystyle PH{P}^{\ast }=A}$$ where${\displaystyle P^{\ast }}$ denotes theconjugate transpose.

A square${\displaystyle n\times n}$ matrix${\displaystyle A}$ is said to be inupper Hessenberg form or to be anupper Hessenberg matrix if${\displaystyle a_{i,j}=0}$ for all${\displaystyle i,j}$ with${\displaystyle i>j+1}$.

An upper Hessenberg matrix is calledunreduced if all subdiagonal entries are nonzero, i.e. if${\displaystyle a_{i+1,i}\ne 0}$ for all${\displaystyle i\in \{1,\dots ,n-1\}}$.^[3]

A square${\displaystyle n\times n}$ matrix${\displaystyle A}$ is said to be inlower Hessenberg form or to be alower Hessenberg matrix if its transpose${\displaystyle }$ is an upper Hessenberg matrix or equivalently if${\displaystyle a_{i,j}=0}$ for all${\displaystyle i,j}$ with${\displaystyle j>i+1}$.

A lower Hessenberg matrix is calledunreduced if all superdiagonal entries are nonzero, i.e. if${\displaystyle a_{i,i+1}\ne 0}$ for all${\displaystyle i\in \{1,\dots ,n-1\}}$.

Consider the following matrices.

The matrix${\displaystyle A}$ is an upper unreduced Hessenberg matrix,${\displaystyle B}$ is a lower unreduced Hessenberg matrix and${\displaystyle C}$ is a lower Hessenberg matrix but is not unreduced.

Many linear algebraalgorithms require significantly lesscomputational effort when applied totriangular matrices, and this improvement often carries over to Hessenberg matrices as well. If the constraints of a linear algebra problem do not allow a general matrix to be conveniently reduced to a triangular one, reduction to Hessenberg form is often the next best thing. In fact, reduction of any matrix to a Hessenberg form can be achieved in a finite number of steps (for example, throughHouseholder's transformation of unitary similarity transforms). Subsequent reduction of Hessenberg matrix to a triangular matrix can be achieved through iterative procedures, such as shiftedQR-factorization. Ineigenvalue algorithms, the Hessenberg matrix can be further reduced to a triangular matrix through Shifted QR-factorization combined with deflation steps. Reducing a general matrix to a Hessenberg matrix and then reducing further to a triangular matrix, instead of directly reducing a general matrix to a triangular matrix, often economizes the arithmetic involved in theQR algorithm for eigenvalue problems.

Any${\displaystyle n\times n}$ matrix can be transformed into a Hessenberg matrix by a similarity transformation usingHouseholder transformations. The following procedure for such a transformation is adapted fromA Second Course In Linear Algebra byGarcia & Horn.^[4]

Let${\displaystyle A}$ be any real or complex${\displaystyle n\times n}$ matrix, then let${\displaystyle A^{\mathrm{\prime }}}$ be the${\displaystyle (n-1)\times n}$ submatrix of${\displaystyle A}$ constructed by removing the first row in${\displaystyle A}$ and let${\displaystyle {\mathbf{a}}_1^{\mathrm{\prime }}}$ be the first column of${\displaystyle A^{\prime }}$. Construct the${\displaystyle (n-1)\times (n-1)}$ householder matrix${\displaystyle V_1=I_{(n-1)}-2\frac{w{w}^{\ast }}{\Vert w{\Vert }^2}}$ where$${\displaystyle w=\{\begin{array}{l}\Vert {\mathbf{a}}_1^{\mathrm{\prime }}{\Vert }_2{\mathbf{e}}_1-{\mathbf{a}}_1^{\mathrm{\prime }},a_{11}^{\mathrm{\prime }}=0\\ \Vert {\mathbf{a}}_1^{\mathrm{\prime }}{\Vert }_2{\mathbf{e}}_1+\frac{\overline{a_{11}^{\mathrm{\prime }}}}{|a_{11}^{\mathrm{\prime }}|}{\mathbf{a}}_1^{\mathrm{\prime }},a_{11}^{\mathrm{\prime }}\ne 0\end{array}}$$

This householder matrix will map${\displaystyle {\mathbf{a}}_1^{\mathrm{\prime }}}$ to${\displaystyle \Vert {\mathbf{a}}_1^{\mathrm{\prime }}\Vert {\mathbf{e}}_1}$ and as such, the block matrix will map the matrix${\displaystyle A}$ to the matrix${\displaystyle U_{1}A}$ which has only zeros below the second entry of the first column. Now construct${\displaystyle (n-2)\times (n-2)}$ householder matrix${\displaystyle V_2}$ in a similar manner as${\displaystyle V_1}$ such that${\displaystyle V_2}$ maps the first column of${\displaystyle A^{\mathrm{\prime }\mathrm{\prime }}}$ to${\displaystyle \Vert {\mathbf{a}}_1^{\mathrm{\prime }\mathrm{\prime }}\Vert {\mathbf{e}}_1}$, where${\displaystyle A^{\mathrm{\prime }\mathrm{\prime }}}$ is the submatrix of${\displaystyle A^{\mathrm{\prime }}}$ constructed by removing the first row and the first column of${\displaystyle A^{\mathrm{\prime }}}$, then let which maps${\displaystyle U_{1}A}$ to the matrix${\displaystyle U_2{U}_{1}A}$ which has only zeros below the first and second entry of the subdiagonal. Now construct${\displaystyle V_3}$ and then${\displaystyle U_3}$ in a similar manner, but for the matrix${\displaystyle A^{\mathrm{\prime }\mathrm{\prime }\mathrm{\prime }}}$ constructed by removing the first row and first column of${\displaystyle A^{\mathrm{\prime }\mathrm{\prime }}}$ and proceed as in the previous steps. Continue like this for a total of${\displaystyle n-2}$ steps.

By construction of${\displaystyle U_k}$, the first${\displaystyle k}$ columns of any${\displaystyle n\times n}$ matrix are invariant under multiplication by${\displaystyle U_k^{\ast }}$ from the right. Hence, any matrix can be transformed to an upper Hessenberg matrix by a similarity transformation of the form${\displaystyle U_{(n-2)}(\dots (U_2(U_{1}A{U}_1^{\ast })U_2^{\ast })\dots )U_{(n-2)}^{\ast }=U_{(n-2)}\dots U_2{U}_{1}A(U_{(n-2)}\dots U_2{U}_1{)}^{\ast }=UA{U}^{\ast }}$.

AJacobi rotation (also called Givens rotation) is an orthogonal matrix transformation in the form

${\displaystyle A\to A^{\prime }=J(p,q,\theta {)}^{T}AJ(p,q,\theta ),}$

where${\displaystyle J(p,q,\theta )}$,, is the Jacobi rotation matrix with all matrix elements equal zero except for

One can zero the matrix element${\displaystyle A_{p-1,q}^{\prime }}$ by choosingthe rotation angle${\displaystyle \theta }$ to satisfy the equation

${\displaystyle A_{p-1,p}\sin \theta +A_{p-1,q}\cos \theta =0,}$

Now, the sequence of such Jacobi rotations with the following${\displaystyle (p,q)}$

${\displaystyle (p,q)=(2,3),(2,4),\dots ,(2,n),(3,4),\dots ,(3,n),\dots ,(n-1,n)}$

reduces the matrix${\displaystyle A}$ to the lower Hessenberg form.^[5]

For${\displaystyle n\in \{1,2\}}$, every${\displaystyle n\times n}$ matrix is both upper Hessenberg, and lower Hessenberg.^[6]

The product of a Hessenberg matrix with a triangular matrix is again Hessenberg. More precisely, if${\displaystyle A}$ is upper Hessenberg and${\displaystyle T}$ is upper triangular, then${\displaystyle AT}$ and${\displaystyle TA}$ are upper Hessenberg.

A matrix that is both upper Hessenberg and lower Hessenberg is atridiagonal matrix, of which theJacobi matrix is an important example. This includes the symmetric or Hermitian Hessenberg matrices. A Hermitian matrix can be reduced to tri-diagonal real symmetric matrices.^[7]

The Hessenberg matrix can be presented in a Jordan canonical form, withconfluent Vandermonde matrix as the similarity matrix (chapter 1.4.2 of^[8]). For review of algorithms inverting this form of similarity matrix, necessary to construct that Jordan form, look into chapter 1.9 also of^[8]. For spectral properties of Hessenberg, and matrix look also into chapter 1.9 of^[8].

The Hessenberg operator is an infinite dimensional Hessenberg matrix. It commonly occurs as the generalization of theJacobi operator to a system oforthogonal polynomials for the space ofsquare-integrableholomorphic functions over some domain—that is, aBergman space. In this case, the Hessenberg operator is the right-shift operator${\displaystyle S}$, given by$${\displaystyle [Sf](z)=zf(z).}$$

Theeigenvalues of each principal submatrix of the Hessenberg operator are given by thecharacteristic polynomial for that submatrix. These polynomials are called theBergman polynomials, and provide anorthogonal polynomial basis for Bergman space.

• Hessenberg variety

• Horn, Roger A.; Johnson, Charles R. (1985),Matrix Analysis,Cambridge University Press,ISBN 978-0-521-38632-6.
• Stoer, Josef; Bulirsch, Roland (2002),Introduction to Numerical Analysis (3rd ed.), Berlin, New York:Springer-Verlag,ISBN 978-0-387-95452-3.
• Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007),"Section 11.6.2. Reduction to Hessenberg Form",Numerical Recipes: The Art of Scientific Computing (3rd ed.), New York: Cambridge University Press,ISBN 978-0-521-88068-8, archived fromthe original on 2011-08-11, retrieved2011-08-13

• Hessenberg matrix atMathWorld.
• Hessenberg matrix atPlanetMath.
• High performance algorithms for reduction to condensed (Hessenberg, tridiagonal, bidiagonal) form
• Algorithm overview

• Matrices (mathematics)

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