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矩陣求導與轉置運算

線性代數補充


前言:在推導演算法過程中遇到一些數學運算,遇到了就記錄下,方便回憶


矩陣轉置

( A + B ) T

= A T + B T ( A
B ) T = B T A T
(A+B)^T = A^T+B^T \\ (AB)^T = B^TA^T

矩陣求導

A x x = A T A x x T = A x T A x = A \frac{\partial Ax}{\partial x} = A^T \\ \frac{\partial Ax}{\partial x^T} = A \\ \frac{\partial x^TA}{\partial x} = A \\
x T A x x = ( A T + A ) x \frac{\partial x^TAx}{\partial x} = (A^T+A)x 若A是對稱矩陣,即A^T = A,則:
x T A x x = 2 A x \frac{\partial x^TAx}{\partial x} = 2Ax


參考:
向量,標量對向量求導數