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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]
Social rank theory provides an evolutionary paradigm that locates affiliative and ranking structures at the core of many psychological disorders. In this context, displays of submission signal to dominant individuals that subordinate group members are not a threat to their rank within the social hierarchy .
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 ...
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} .
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.
In mathematics and multivariate statistics, the centering matrix [1] is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component of that vector.
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 ...