It can be seen that, since the entropy production must be greater than zero, the Onsager matrix of phenomenological coefficients L _ { \ alpha \ beta } is a positive semi-definite matrix.
22.
An important class of matrix-valued reproducing kernels are " separable " kernels which can factorized as the product of a scalar valued kernel and a T-dimensional symmetric positive semi-definite matrix.
23.
Where \ mathbf { x } is a real non-zero column-vector and A is a real symmetric positive-definite matrix, is the smallest eigenvalue of A, while the minimizer is the corresponding eigenvector.
24.
Standard direct methods, i . e ., methods that use some matrix decomposition are Gaussian elimination, LU decomposition, Cholesky decomposition for hermitian ) and positive-definite matrix, and QR decomposition for non-square matrices.
25.
Where Z is a normalization constant, " A " is a symmetric positive definite matrix ( inverse covariance matrix a . k . a . precision matrix ) and " b " is the shift vector.
26.
Moreover, it is a strictly positive-definite matrix, so that the system " AU " = " F " always has a unique solution . ( For other problems, these nice properties will be lost .)
27.
Where \ mathbf { x } and \ mathbf { b } are real column-vectors and A is a real symmetric positive-definite matrix, is exactly the solution of the linear equation A \ mathbf { x } = \ mathbf { b }.
28.
The matrix \ mathbf X ^ { \ rm T } \ mathbf X is known as the Gramian matrix of \ mathbf X, which possesses several nice properties such as being a positive semi-definite matrix, and the matrix \ mathbf X ^ { \ rm T } \ mathbf y is known as the moment matrix of regressand by regressors.
29.
This also leads to a proof of the above observation, that a positive-definite matrix has precisely one positive-definite square root : a positive definite matrix has only positive eigenvalues, and each of these eigenvalues has only one positive square root; and since the eigenvalues of the square root matrix are the diagonal elements of ? for the square root matrix to be itself positive definite necessitates the use of only the unique positive square roots of the original eigenvalues.
30.
This also leads to a proof of the above observation, that a positive-definite matrix has precisely one positive-definite square root : a positive definite matrix has only positive eigenvalues, and each of these eigenvalues has only one positive square root; and since the eigenvalues of the square root matrix are the diagonal elements of ? for the square root matrix to be itself positive definite necessitates the use of only the unique positive square roots of the original eigenvalues.
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