关于对称矩阵随机变换方差的证明

Proof for Variance of Random Variable after Stochastic Transformation of Symmetric Matrix

应用正交变换将对称矩阵对角化,基于随机向量正交变换后独立性的不变性及矩阵迹相关性质,给出一个关于对称矩阵经随机变换后方差的证明,并将该结论推广到更一般情形。

Based on the Theorem in a reference book, a symmetric matrix is diagonalized by orthogonal transformation first. Because of the invariance property of independence when a random variable is transformed through orthogonal mean,a proof for variance of random variable after stochastic transformation of symmetric matrix is presented by applying the property of trace of matrix. Secondly,a general conclusion for the theorem is studied.