Suppose T is positive-definite Hermitian matrix and I know that it can be expressed by eigen-decomposition as the following sum of rank-1 matrices:$ \textbf{T}= \sum \lambda _{k} \textbf{u}_{k} \textbf{u}_{k}^{H} $where $\textbf{u}_{k} $ are orthogonal to each other.
But my question is: is this rank-1 decomposition unique? For example, can T be also written in other forms, say:$\textbf{T}= \sum \gamma _{k} \textbf{v}_{k} \textbf{v}_{k}^{H} $, only in this case $\textbf{v}_{k} $ do not necessarily need to be orthogonal vectors. If so, is there any relationship between $\textbf{u}_{k} $ and $\textbf{v}_{k} $?
Thanks.
Best Answer
Nope. For example (exercise!), the identity matrix is equal to $\sum \textbf{u}_k \textbf{u}_k^H$ for every orthonormal basis $\textbf{u}_k$.