Hankel matrix

In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.: In linear algebra, a Hankel matrix (or catalecticant matrix), named after Hermann Hankel, is a square matrix in which each ascending skew-diagonal from left to right is constant, e.g.: More generally, a Hankel matrix is any n × n {displaystyle n imes n} matrix A {displaystyle A} of the form A = [ a 0 a 1 a 2 … … a n − 1 a 1 a 2 ⋮ a 2 ⋮ ⋮ a 2 n − 4 ⋮ a 2 n − 4 a 2 n − 3 a n − 1 … … a 2 n − 4 a 2 n − 3 a 2 n − 2 ] . {displaystyle A={egin{bmatrix}a_{0}&a_{1}&a_{2}&ldots &ldots &a_{n-1}\a_{1}&a_{2}&&&&vdots \a_{2}&&&&&vdots \vdots &&&&&a_{2n-4}\vdots &&&&a_{2n-4}&a_{2n-3}\a_{n-1}&ldots &ldots &a_{2n-4}&a_{2n-3}&a_{2n-2}end{bmatrix}}.} In terms of the components, if the i , j {displaystyle i,j} element of A {displaystyle A} is denoted with A i j {displaystyle A_{ij}} , and assuming i ≤ j {displaystyle ileq j} , then we have The Hankel matrix is a symmetric matrix. The Hankel matrix is closely related to the Toeplitz matrix (a Hankel matrix is an upside-down Toeplitz matrix). For a special case of this matrix see Hilbert matrix. A Hankel operator on a Hilbert space is one whose matrix with respect to an orthonormal basis is a (possibly infinite) Hankel matrix ( A i , j ) i , j ≥ 1 {displaystyle (A_{i,j})_{i,jgeq 1}} , where A i , j {displaystyle A_{i,j}} depends only on i + j {displaystyle i+j} . The determinant of a Hankel matrix is called a catalecticant. The Hankel transform is the name sometimes given to the transformation of a sequence, where the transformed sequence corresponds to the determinant of the Hankel matrix. That is, the sequence { h n } n ≥ 0 {displaystyle {h_{n}}_{ngeq 0}} is the Hankel transform of the sequence { b n } n ≥ 0 {displaystyle {b_{n}}_{ngeq 0}} when