Today in the class Prof. applied spectral theorem and wrote $A$ a semidefinite positive matrix as $A=\sum \lambda_i v_i\times v_i$ , where $v_i$ are the eigenvectores and $\lambda_i$ are corresponding eigen values. I think $"\times"$ should be some kind of tensor product or so ( i don't know ) . I don't know spectral theorem. Any kind of explanation would be nice .
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It's the outer product: writing $v_i$ as a column you have $v_i v_i^T$. The spectral theorem says that a self-adjoint matrix (over $\mathbb{R}$, a symmetric matrix) admits a basis of orthonormal eigenvectors $v_j$: $Av_j = \lambda_j v_j$. You can easily show that $(\sum \lambda_j v_j v_j^T) v_i = \lambda_i v_i$. Since the two matrices $A$ and $\sum \lambda_j v_j v_j^T$ map the basis $v_i$ to the same vectors, they are equal. |
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If $v_i$ is a column vector ($N$ rows), one column, then $v_i\times v_i=v_iv_i^T$ where $v_i^T$ is the transposed vector and the product is the usual one. |
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