In linear algebra, a Hessenberg matrix is a special kind of square matrix, one that is "almost" triangular. To be exact, an upper Hessenberg matrix has zero entries below the first subdiagonal, and a lower Hessenberg matrix has zero entries above the first superdiagonal. They are named after Karl Hessenberg. A Hessenberg decomposition is a matrix decomposition of a matrix A {\displaystyle A} into a unitary matrix P {\displaystyle P} and a Hessenberg matrix H {\displaystyle H} such that P H P ∗ = A {\displaystyle PHP^{*}=A} where P ∗ {\displaystyle P^{*}} denotes the conjugate transpose.
Definitions
Upper Hessenberg matrix A square n × n {\displaystyle n\times n} matrix A {\displaystyle A} is said to be in upper Hessenberg form or to be an upper Hessenberg matrix if a i , j = 0 {\displaystyle a_{i,j}=0} for all i , j {\displaystyle i,j} with i > j + 1 {\displaystyle i>j+1} . In R n × n , {\displaystyle \mathbb {R} ^{n\times n},} it has the following structure:
A = ( a 11 a 12 a 13 ⋯ a 1 n a 21 a 22 a 23 ⋯ a 2 n 0 a 32 a 33 ⋯ a 3 n ⋮ ⋱ ⋱ ⋱ ⋮ 0 ⋯ 0 a n , n − 1 a n n ) . {\displaystyle A={\begin{pmatrix}a_{11}&a_{12}&a_{13}&\cdots &a_{1n}\\a_{21}&a_{22}&a_{23}&\cdots &a_{2n}\\0&a_{32}&a_{33}&\cdots &a_{3n}\\\vdots &\ddots &\ddots &\ddots &\vdots \\0&\cdots &0&a_{n,n-1}&a_{nn}\end{pmatrix}}.}
An upper Hessenberg matrix is called unreduced if all subdiagonal entries are nonzero, i.e. if a i + 1 , i ≠ 0 {\displaystyle a_{i+1,i}\neq 0} for all i ∈ { 1 , … , n − 1 } {\displaystyle i\in \{1,\ldots ,n-1\}} .
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