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Any square matrix can uniquely be written as sum of a symmetric and a skew-symmetric matrix. This decomposition is known as the Toeplitz decomposition. Let Mat n {\displaystyle {\mbox{Mat}}_{n}} denote the space of n × n {\displaystyle n\times n} matrices.
The minimum rank of a graph is always at most equal to n − 1, where n is the number of vertices in the graph. [1] For every induced subgraph H of a given graph G, the minimum rank of H is at most equal to the minimum rank of G. [2] If a graph is disconnected, then its minimum rank is the sum of the minimum ranks of its connected components. [3]
A matrix that has rank min(m, n) is said to have full rank; otherwise, the matrix is rank deficient. Only a zero matrix has rank zero. f is injective (or "one-to-one") if and only if A has rank n (in this case, we say that A has full column rank). f is surjective (or "onto") if and only if A has rank m (in this case, we say that A has full row ...
In mathematics, a symmetric matrix with real entries is positive-definite if the real number is positive for every nonzero real column vector, where is the row vector transpose of . [1] More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number is positive for every nonzero complex column vector , where denotes the ...
Every finite-dimensional matrix has a rank decomposition: Let be an matrix whose column rank is . Therefore, there are r {\textstyle r} linearly independent columns in A {\textstyle A} ; equivalently, the dimension of the column space of A {\textstyle A} is r {\textstyle r} .
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
In matrix theory and combinatorics, a Pascal matrix is a matrix (possibly infinite) containing the binomial coefficients as its elements. It is thus an encoding of Pascal's triangle in matrix form. There are three natural ways to achieve this: as a lower-triangular matrix, an upper-triangular matrix, or a symmetric matrix. For example, the 5 × ...
are also possible. The minimum number r for which such a decomposition is possible is the symmetric rank of T. [3] This minimal decomposition is called a Waring decomposition; it is a symmetric form of the tensor rank decomposition. For second-order tensors this corresponds to the rank of the matrix representing the tensor in any basis, and it ...