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If I take $n$ linearly independent vectors ($n \times 1$), $v_1, v_2, \ldots, v_n$ and construct a matrix which is the sum of their outer products,

$$M = \sum_i^n v_iv_i^t$$

Then, can it be proven that $M$ is always full rank?

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    $\begingroup$ Such a matrix will be positive definite. $\endgroup$ Commented Aug 25, 2017 at 22:57
  • $\begingroup$ Can we prove this? $\endgroup$ Commented Aug 25, 2017 at 23:01
  • $\begingroup$ ^Exploit the fact that the $\{v_i\}$ form a basis. $\endgroup$ Commented Aug 25, 2017 at 23:15

2 Answers 2

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If $M = \sum v_i v_{i}^{T}$ then

\begin{align} Mx & = \sum v_i v_i^T x \\[10pt] & = \sum v_i (v_i^T x) \\[10pt] & = \sum (c_i)v_i \end{align}

where $c_i$ is a scalar denoting the inner product of $x$ and $v_i$. Since the $v_i's$ are linearly independent this sum yields the zero vector if and only if $c_i=0$ for all $i$ and hence $x$ is the zero vector. So your matrix clearly has full rank.

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    $\begingroup$ Just wanted to add the proof is using the rank nullity theorem. Also the vectors $v_i$ in the sum don't have to be identical. It is enough that they are linearly independent. $\endgroup$
    – KFkf
    Commented May 17, 2021 at 9:05
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First, notice that $M$ being full-rank is equivalent to the induced linear operator being injective, which is equivalent to $\det M \neq 0$.

Now define $T : e_i \mapsto v_i$, where $e_i$ is the canonical basis of $\mathbb R^n$. Then $v_i^T = (T e_i)^T = e_i^T T^T$, hence $$M = \sum_i T e_i e_i^T T^T = T \left( \sum_i e_i e_i^T \right) T^T$$ But the matrix in brackets is just the identity matrix, thus $$M = T T^T \quad \Longrightarrow \quad \det M = ( \det T)^2 $$ But $T$ is full rank, thus $\det T \neq 0$ and $\det M \neq 0$.

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  • $\begingroup$ But the existence of such a $T$ is contingent on $M$ being invertible, no? $\endgroup$ Commented Aug 25, 2017 at 23:16

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