Linear Algebra

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

Problem 640

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

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

Add to solve later

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 $1 \leq i, j \leq n$.

Proof.

Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix}$. Then we have
\begin{align*}
\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
\end{bmatrix}=
\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}.
\end{align*}

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.

Add to solve later

More from my site

The Length of a Vector is Zero if and only if the Vector is the Zero Vector Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$. Proof. Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} $. Then we […]
Subspaces of Symmetric, Skew-Symmetric Matrices Let $V$ be the vector space over $\R$ consisting of all $n\times n$ real matrices for some fixed integer $n$. Prove or disprove that the following subsets of $V$ are subspaces of $V$. (a) The set $S$ consisting of all $n\times n$ symmetric matrices. (b) The set $T$ consisting of […]
Diagonalizable by an Orthogonal Matrix Implies a Symmetric Matrix Let $A$ be an $n\times n$ matrix with real number entries. Show that if $A$ is diagonalizable by an orthogonal matrix, then $A$ is a symmetric matrix. Proof. Suppose that the matrix $A$ is diagonalizable by an orthogonal matrix $Q$. The orthogonality of the […]
Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by \[A=I-a\mathbf{v}\mathbf{v}^{\trans},\] where […]
Transpose of a Matrix and Eigenvalues and Related Questions Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ […]
$If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix$ If $A^{\trans}A=A$, then $A$ is a Symmetric Idempotent Matrix Let $A$ be a square matrix such that \[A^{\trans}A=A,\] where $A^{\trans}$ is the transpose matrix of $A$. Prove that $A$ is idempotent, that is, $A^2=A$. Also, prove that $A$ is a symmetric matrix. Hint. Recall the basic properties of transpose […]
Inverse Matrix of Positive-Definite Symmetric Matrix is Positive-Definite Suppose $A$ is a positive definite symmetric $n\times n$ matrix. (a) Prove that $A$ is invertible. (b) Prove that $A^{-1}$ is symmetric. (c) Prove that $A^{-1}$ is positive-definite. (MIT, Linear Algebra Exam Problem) Proof. (a) Prove that $A$ is […]
$Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$$ Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$ Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors. (a) Prove that $\mathbf{v}^\trans \mathbf{w} = \mathbf{w}^\trans \mathbf{v}$. (b) Provide an example to show that $\mathbf{v} \mathbf{w}^\trans$ is not always equal to $\mathbf{w} […]