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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]
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 ...
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} .
Hat matrix — a square matrix used in statistics to relate fitted values to observed values. Orthostochastic matrix — doubly stochastic matrix whose entries are the squares of the absolute values of the entries of some orthogonal matrix; Precision matrix — a symmetric n×n matrix, formed by inverting the covariance 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 × ...