Search results
Results From The WOW.Com Content Network
A complex symmetric matrix can be 'diagonalized' using a unitary matrix: thus if is a complex symmetric matrix, there is a unitary matrix such that is a real diagonal matrix with non-negative entries.
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 ...
If instead, A is equal to the negative of its transpose, that is, A = −A T, then A is a skew-symmetric matrix. In complex matrices, symmetry is often replaced by the concept of Hermitian matrices, which satisfies A ∗ = A, where the star or asterisk denotes the conjugate transpose of the matrix, that is, the transpose of the complex ...
There is a dual notion of co-rank of a finitely generated group G defined as the largest cardinality of X such that there exists an onto homomorphism G → F(X). Unlike rank, co-rank is always algorithmically computable for finitely presented groups, [17] using the algorithm of Makanin and Razborov for solving systems of equations in free groups.
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
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 ...
A logically equivalent definition is given by Rudolf & Woeginger who in 1995 proved that A matrix is a Supnick matrix iff it can be written as the sum of a sum matrix S and a non-negative linear combination of LL-UR block matrices. The sum matrix is defined in terms of a sequence of n real numbers {α i}: