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Lets say we have a symmetric positive semi-definite $n\times n$ matrix $\Sigma$, which therefore has a diagonalization $\Sigma=V\Lambda V^T$, where $V$ is an orthogonal matrix (containing the eigenvectors $v_1,...v_n$ of $\Sigma$) and $\Lambda$ is a diagonal matrix (containing the eigenvalues $\lambda_1,...,\lambda_n$ of $\Sigma$).

Could you show me how the following sum of outer products is derived?

$\Sigma=\sum_{i=1}^n \lambda_iv_iv_i^T$

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It helps to have internalized the following ways of looking at matrix multiplication:

  • $Ax = a_1 x_1 + \cdots + a_n x_n$, where $a_i$ is the $i$th column of the matrix $A$ and $x_i$ is the $i$th component of the vector $x$.
  • $A \begin{bmatrix} b_1 & \cdots & b_n \end{bmatrix} = \begin{bmatrix} A b_1 & \cdots & A b_n \end{bmatrix}$, where $b_i$ is the $i$th column of the matrix $B$.
  • $\begin{bmatrix} a_1 & \cdots & a_n \end{bmatrix} \begin{bmatrix} b_1^T \\ \vdots \\ b_n^T \end{bmatrix} = a_1 b_1^T + \cdots + a_n b_n^T$. Here the vector $a_i$ is the $i$th column of $A$ and the row vector $b_i^T$ is the $i$th row of $B$.

You can use rules 2 and 1 to compute $V \Lambda$, and then use rule 3 to compute $V \Lambda V^T$.

Alternatively, you can think of it like this. The matrix $v_i v_i^T$ projects a vector $x$ onto the span of $\{ v_i \}$. This projection then gets scaled by $\lambda_i$. In other words: \begin{equation} x = \sum_{i=1}^n v_i v_i^T x \end{equation} so \begin{align} \Sigma x &= \sum_{i=1}^n \Sigma v_i v_i^T x \\ &= \sum_{i=1}^n \lambda_i v_i v_i^T x. \end{align}

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  • $\begingroup$ Thank you very much! I've actually never thought of rule 3 - that one clarified everything for me. $\endgroup$ – ndrizza Jul 12 '16 at 6:59
  • $\begingroup$ Sure. Gilbert Strang emphasizes rule 3 (as well as the other viewpoints) in his linear algebra books. $\endgroup$ – littleO Jul 12 '16 at 7:05
  • $\begingroup$ Indeed his book is great. Unfortunately, I only only discovered it four days before my exam, so I only read selected chapters (which were great tough). I wish I had discovered his book before. $\endgroup$ – ndrizza Jul 12 '16 at 7:19

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