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Gramian matrix: The symmetric matrix of the pairwise inner products of a set of vectors in an inner product space: Hessian matrix: The square matrix of second partial derivatives of a function of several variables: Householder matrix: The matrix of a reflection with respect to a hyperplane passing through the origin: Jacobian matrix
If the matrix is symmetric indefinite, it may be still decomposed as = where is a permutation matrix (arising from the need to pivot), a lower unit triangular matrix, and is a direct sum of symmetric and blocks, which is called Bunch–Kaufman decomposition [6]
SageMath is designed partially as a free alternative to the general-purpose mathematics products Maple and MATLAB. It can be downloaded or used through a web site. SageMath comprises a variety of other free packages, with a common interface and language. SageMath is developed in Python.
A symmetric arrowhead matrix is irreducible if for all i and for all . The eigenvalues of an irreducible real symmetric arrowhead matrix are the zeros of the secular equation f ( λ ) = α − λ − ∑ i = 1 n − 1 ζ i 2 d i − λ ≡ α − λ − z T ( D − λ I ) − 1 z = 0 {\displaystyle f(\lambda )=\alpha -\lambda -\sum _{i=1}^{n-1 ...
Download as PDF; Printable version; ... In mathematics, a Hamiltonian matrix is a 2n-by-2n matrix A such that JA is symmetric, where J is the skew-symmetric matrix = ...
In mathematics, persymmetric matrix may refer to: a square matrix which is symmetric with respect to the northeast-to-southwest diagonal (anti-diagonal); or; a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line. The first definition is the most common in the recent literature.
ML.NET is a free-software machine-learning library for the C# programming language. [4] [5] NAG Library is an extensive software library of highly optimized numerical-analysis routines for various programming environments. O-Matrix is a proprietary licensed matrix programming language for mathematics, engineering, science, and financial analysis.
If the characteristic of the field is 2, then a skew-symmetric matrix is the same thing as a symmetric matrix. The sum of two skew-symmetric matrices is skew-symmetric. A scalar multiple of a skew-symmetric matrix is skew-symmetric. The elements on the diagonal of a skew-symmetric matrix are zero, and therefore its trace equals zero.