Jacobi’s formula for the derivative of a determinant
gotohaggstrom.com › Jacobis formula for theTo calculate the derivative of the determinant we use (13) with: f(A) = f(a 11;a 12;a 21;a 22) = a 11a 22 a 21a 12 (18) Thus: @f @a 11 =a 22 @f @a 12 = a 21 @f @a 21 = a 12 @f @a 22 =a 11 (19) Also: da 11 dt =2 da 12 dt =1 da 21 dt = 1 da 22 dt =3 (20) So formula (13) gives: df dt = X2 i=1 X2 j=1 @f @a ij da ij dt = 3t 2 + t 1 + ( t) ( 1) + 2t 3 = 14t (21) From (14) d dt A = 14t. Thus con rming (1): jAj jAj = 14t 7t2 =
Jacobi's formula - Wikipedia
https://en.wikipedia.org/wiki/Jacobi's_formulaIn matrix calculus, Jacobi's formula expresses the derivative of the determinant of a matrix A in terms of the adjugate of A and the derivative of A.. If A is a differentiable map from the real numbers to n × n matrices, then = ( (()) ()) = (()) (() ())where tr(X) is the trace of the matrix X.. As a special case, = ().Equivalently, if dA stands for the differential of A, the ...
Log-Determinant Function and Properties
inst.eecs.berkeley.edu › def_logdet_fcnApr 08, 2021 · Log-Determinant Function and Properties. The log-determinant function is a function from the set of symmetric matrices in Rn×n R n × n, with domain the set of positive definite matrices, and with values. f (X)= {logdetX if X ≻ 0, +∞ otherwise. f ( X) = { log. . det X if X ≻ 0, + ∞ otherwise. The function can be expressed in terms of the (real, positive) eigenvalues of the argument matrix X X; it does not depend on its eigenvectors.