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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 ...
In mathematics, negative definiteness is a property of any object to which a bilinear form may be naturally associated, which is negative-definite. See, in particular: Negative-definite bilinear form; Negative-definite quadratic form; Negative-definite matrix; Negative-definite function
In the matrix notation, the adjacency matrix of the undirected graph could, e.g., be defined as a Boolean sum of the adjacency matrix of the original directed graph and its matrix transpose, where the zero and one entries of are treated as logical, rather than numerical, values, as in the following example:
The Hessian matrix plays an important role in Morse theory and catastrophe theory, because its kernel and eigenvalues allow classification of the critical points. [2] [3] [4] The determinant of the Hessian matrix, when evaluated at a critical point of a function, is equal to the Gaussian curvature of the function considered as a manifold. The ...
Suppose a vector norm ‖ ‖ on and a vector norm ‖ ‖ on are given. Any matrix A induces a linear operator from to with respect to the standard basis, and one defines the corresponding induced norm or operator norm or subordinate norm on the space of all matrices as follows: ‖ ‖, = {‖ ‖: ‖ ‖ =} = {‖ ‖ ‖ ‖:} . where denotes the supremum.
Non-negative matrix factorization (NMF or NNMF), also non-negative matrix approximation [1] [2] is a group of algorithms in multivariate analysis and linear algebra where a matrix V is factorized into (usually) two matrices W and H, with the property that all three matrices have no negative elements. This non-negativity makes the resulting ...
where is a finite matrix, is asymptotically stable (in fact, exponentially stable) if all real parts of the eigenvalues of are negative. This condition is equivalent to the following one: [ 12 ] A T M + M A {\displaystyle A^{\textsf {T}}M+MA}
The exponential representation of an orthogonal matrix of order can also be obtained starting from the fact that in dimension any special orthogonal matrix can be written as =, where is orthogonal and S is a block diagonal matrix with ⌊ / ⌋ blocks of order 2, plus one of order 1 if is odd; since each single block of order 2 is also an ...