Matrix Operations with Transpose

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• 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} […]
• A Relation between the Dot Product and the Trace Let $\mathbf{v}$ and $\mathbf{w}$ be two $n \times 1$ column vectors. Prove that $\tr ( \mathbf{v} \mathbf{w}^\trans ) = \mathbf{v}^\trans \mathbf{w}$.   Solution. Suppose the vectors have components \[\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n […]
• Is the Trace of the Transposed Matrix the Same as the Trace of the Matrix? 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.   Solution. The answer is true. Recall that the transpose of a matrix is the sum of its diagonal entries. Also, note that the […]
• Rotation Matrix in Space and its Determinant and Eigenvalues For a real number $0\leq \theta \leq \pi$, we define the real $3\times 3$ matrix $A$ by \[A=\begin{bmatrix} \cos\theta & -\sin\theta & 0 \\ \sin\theta &\cos\theta &0 \\ 0 & 0 & 1 \end{bmatrix}.\] (a) Find the determinant of the matrix $A$. (b) Show that $A$ is an […]
• Column Rank = Row Rank. (The Rank of a Matrix is the Same as the Rank of its Transpose) Let $A$ be an $m\times n$ matrix. Prove that the rank of $A$ is the same as the rank of the transpose matrix $A^{\trans}$.   Hint. Recall that the rank of a matrix $A$ is the dimension of the range of $A$. The range of $A$ is spanned by the column vectors of the matrix […]
• If $A$ is an Idempotent Matrix, then When $I-kA$ is an Idempotent Matrix? A square matrix $A$ is called idempotent if $A^2=A$. (a) Suppose $A$ is an $n \times n$ idempotent matrix and let $I$ be the $n\times n$ identity matrix. Prove that the matrix $I-A$ is an idempotent matrix. (b) Assume that $A$ is an $n\times n$ nonzero idempotent matrix. Then […]
• Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let $A, B, C$ be the following $3\times 3$ matrices. \[A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […]
• Find the Distance Between Two Vectors if the Lengths and the Dot Product are Given Let $\mathbf{a}$ and $\mathbf{b}$ be vectors in $\R^n$ such that their length are \[\|\mathbf{a}\|=\|\mathbf{b}\|=1\] and the inner product \[\mathbf{a}\cdot \mathbf{b}=\mathbf{a}^{\trans}\mathbf{b}=-\frac{1}{2}.\] Then determine the length $\|\mathbf{a}-\mathbf{b}\|$. (Note […]