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 later Tagged: transpose of a matrix
Add to solve later Prove that the Dot Product is Commutative: $\mathbf{v}\cdot \mathbf{w}= \mathbf{w} \cdot \mathbf{v}$
Problem 637
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 $\mathbf{w} \mathbf{v}^\trans$.
Add to solve later Matrix Operations with Transpose
Problem 636
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) $\mathbf{v}^\trans \mathbf{v} + \mathbf{v}^\trans B A^\trans \mathbf{v}$.
Add to solve later Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix?
Problem 633
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.
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