## A Relation between the Dot Product and the Trace

## Problem 638

Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.

Prove that $\tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans \mathbf{w}$.

Add to solve laterLet $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors.

Prove that $\tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans \mathbf{w}$.

Add to solve laterLet $\mathbf{v}$ 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 $\mathbf{w} \mathbf{v}^\trans$.

Calculate the following expressions, using the following matrices:

\[A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}\]

**(a)** $A B^\trans + \mathbf{v} \mathbf{v}^\trans$.

**(b)** $A \mathbf{v} – 2 \mathbf{v}$.

**(c)** $\mathbf{v}^{\trans} B$.

**(d)** $\mathbf{v}^\trans \mathbf{v} + \mathbf{v}^\trans B A^\trans \mathbf{v}$.

Let $A$ be an $n \times n$ matrix.

Is it true that $\tr ( A^\trans ) = \tr(A)$? If it is true, prove it. If not, give a counterexample.

Add to solve later