Symmetric positive semidefinite matrix

Pearson s linear correlation coefficient, 239 perturbation analysis, pairwise decomposition scheme Frobenius norm, 235 RTB Hessian matrix, 234 symmetric positive semidefinite (SPSD) matrix, 237... 

The difficulty here is how to simultaneously constrain y and / y to be positive semidefinite. To formulate it as a primal SDP problem (Eq. (1)), we should express these two conditions as a positive semidefinite constraint over a single matrix let y be a block-diagonal matrix in which two symmetric matrices yj and y2 are arranged diagonally, and let us express the interrelation between these two matrices via linear constraints defined by the matrices Ap and the constants as in Eq. (1). That is. 

Since ATA is symmetric and positive semidefinite, x is real and nonnegative. If the matrix is nonsingular (and hence positive definite) then... 

A symmetric matrix A is said to be positive-definite if the quadratic form uTAu > 0 for all nonzero vectors u. Similarly, the symmetric matrix A is positive-semidefinite if uTAu 2 0 for all nonzero vectors u. Positive-definite matrices have strictly positive eigenvalues. We classify A as negative-definite if u Au < 0 for all nonzero vectors u. A is indefinite if uTAu is positive for some u and negative for others. 

A symmetric matrix A is said to be positive definite if all of the eigenvalues are positive it is said to be negative definite if all of the eigenvalues are negative. Semidefinite is similarly defined. There is a simple test to determine if a symmetric matrix is positive or negative definite. 

A symmetric matrix is positive semidefinite if all its eigenvalues are nonnegative. A positive definite matrix is one where all its eigenvalues are strictly positive and nonzero. 

Higham, N. J., Computing a nearest symmetric positive semidefinite matrix. Linear Algebra Applic., 103, 103 (1988). 

The verbal interpretation of (5.5) is that the process is continuous — this is the Lindeberg condition. The function a x, t) is the velocity of conditional expectation ( drift vector ), and bjj(x, t) is the matrix of the velocity of conditional covariance ( diffusion matrix ). The latter is positive semidefinite and symmetric as a result of its definition (5.7). 

In Eq. (7.5), (x) and S(x) are the coarse-grained energy and entropy functions, respectively. The antisymmetric operator L defines a generalized Poisson bracket A, = L that possesses the same properties as the classical Poisson bracket described above. The last term in Eq. (7.5) is new compared to Hamiltonian dynamics (7.4) and describes dissipative, irreversible phenomena. The friction matrix M is symmetric and positive, semidefinite. Together with the degeneracy... 

Updated values of the Lagrange multiplier X are calculated from the stationary GENERIC equation(7.23) by inverting the symmetric, positive semidefinite matrix M. 

Thus, a real symmetric matrix A is positive-definite if all of its eigenvalues are positive. If we have only that v Av > 0, is said to be positive-semidefinite. If v Av < 0 or < 0, A is negative-definite or negative-semidefinite respectively. If no such condition holds for all u e A is indefinite. 

In fact, such an extension exists. For a. MxN real matrix A, we can always generate the NxN square matrix A A tiiat is symmetric and positive-semidefinite i.e., all eigenvalues are real and are greater than or equal to zero. Let the eigenvalues of be fx.2,..., fXN- We then define the singular values of as... 

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