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Classification task
Published in Benny Raphael, Construction and Building Automation, 2023
A positive semi-definite matrix is one whose eigenvalues are non-negative. The requirement that the kernel matrix should be positive semi-definite comes from a mathematical theorem called Mercer’s theorem. This condition is essential for the convergence of the solutions.
P
Published in Philip A. Laplante, Comprehensive Dictionary of Electrical Engineering, 2018
positive real (PR) function a rational function H (s) of the complex variable s = + is said to be positive real (PR) if it satisfies the following properties: (1) H (s) is a real number whenever s is a real number, and (2) Re[H (s)] 0 whenever Re[s] > 0, where Re[·] represents the real part of [·]. positive semidefinite a scalar function V (x, t) with continuous partial derivatives with respect to all of its arguments is said to be positive semidefinite if (1) V (0, t) = 0. (2) V (x, t) 0 whenever x = 0. positive semi-definite matrix a symmetric matrix A such that x T Ax >= 0 for any vector x. The eigenvalues of a positive semi-definite matrix are all greater than or equal to zero. positive sequence the set of balanced normal (abc) sequence components used in symmetrical components. Balanced load currents, for example, are strictly positive sequence. positive transition angle the angular portion of the time-based output signal (in degrees) that has a positive slope. This quantity could be loosely interpreted as the "leading edge" angle. positive-sequence reactance the inductive reactance offered by a circuit to the flow of positivesequence currents alone. The positive-sequence reactance is a function of the operating frequency of the circuit and the inductance of the circuit to positive-sequence currents. positivity a system H : X X where x, H x 0 x X
Symmetric Matrices
Published in James K. Peterson, Basic Analysis II, 2020
A similar argument holds if we have determinant of A > 0 but a < 0. The determinant condition will then force d < 0 too. We find that xT Ax < 0. In this case, we say the matrix is negative definite. The eigenvalues are still r=(a+d)±(a−d)2+4b22. But now, since ad − b2> 0, the term (a − d)2+ 4b2= a2− 2ad + d2+ 4b2< a2− 2ad + 4ad + d2= |a + d|2. Since a and d are negative, a + d < 0 and so the second root is always negative. The first root’s sign is determined by (a+d)+|(a−d)2+4b2<(a+d)+|a+d|=0. So both eigenvalues are negative. We have found the matrix A is negative definite if a and d are negative and the determinant of A > 0. Note a negative definite matrix has negative eigenvalues.
An improved SRC method based on virtual samples for face recognition
Published in Journal of Modern Optics, 2018
Lijun Fu, Deyun Chen, Kezheng Lin, Ao Li
Therefore, let . First, we assume that coefficients vector α is fixed. The first-order derivative of is . is the second-order derivative of , where is a unit matrix of size n × n. is a symmetric matrix of size n × n. All principal minors of are greater than or equal to zero, hence is positive semi-definite matrix, i.e. . We can draw a conclusion that is convex if α is fixed. Likewise, we assume that coefficients vector β is fixed. The first-order derivative of is . is the second-order derivative of , where is a unit matrix of size . is a symmetric matrix of size . All principal minors of are greater than or equal to zero, hence is positive semi-definite matrix, i.e. . We can draw a conclusion that is convex if β is fixed.