On vector Karhunen-Loeve transforms and optimal vector transforms

We first prove that the vector Karhunen-Loeve (VKL) transform for any finite many vector-valued signal, x₁,···,x_L, exists. The VKL transform is equivalent to the scalar KL transform for the scalar-valued signal X=(x₁^T,···,x_L ^T)^T. Based on VKL transforms, we provide a necessary and sufficient condition for the existence of the optimal vector transforms (unitary). With the condition, one can see that the optimal unitary vector transforms do not exist in most cases, and therefore needs to use suboptimal unitary vector transforms. We then prove that the optimal nonunitary vector transform for x₁,···,x_L exists when all eigenvalues of the correlation matrix of the signal X are nonzero. We formulate the optimal vector transforms via the VKL transforms