Prove that $\mathbf{v} \mathbf{v}^\trans$ is a Symmetric Matrix for any Vector $\mathbf{v}$

symmetric matrices problems

Problem 640

Let $\mathbf{v}$ be an $n \times 1$ column vector.

Prove that $\mathbf{v} \mathbf{v}^\trans$ is a symmetric matrix.

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Definition (Symmetric Matrix).

A matrix $A$ is called symmetric if $A^{\trans}=A$.

In terms of entries, an $n\times n$ matrix $A=(a_{ij})$ is symmetric if $a_{ij}=a_{ji}$ for all $1 \leq i, j \leq n$.


Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$. Then we have
\mathbf{v} \mathbf{v}^\trans = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}\begin{bmatrix}
v_1 & v_2 & \cdots & v_n
\begin{bmatrix} v_1 v_1 & v_1 v_2 & \cdots & v_1 v_n \\ v_2 v_1 & v_2 v_2 & \cdots & v_2 v_n \\ \vdots & \vdots & \vdots & \vdots \\ v_n v_1 & v_n v_2 & \cdots & v_n v_n \end{bmatrix}.

In particular, the the $(i, j)$-th component is
\[(\mathbf{v} \mathbf{v}^\trans)_{i j} = v_i v_j = v_j v_i = (\mathbf{v} \mathbf{v}^\trans)_{j i}.\] This shows that the matrix $\mathbf{v} \mathbf{v}^\trans$ is symmetric.

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