## 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 laterof the day

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

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

Add to solve laterLet $\mathbf{v}$ 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 $\mathbf{0}$.

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 laterLet $A$ be an $n\times n$ nonsingular matrix.

Prove that the transpose matrix $A^{\trans}$ is also nonsingular.

Add to solve later 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 $I$ is the $n\times n$ identity matrix.

Prove that $A$ is a symmetric matrix and $AA=I$.

Conclude that the inverse matrix is $A^{-1}=A$.

Let $A$ be a square matrix.

Prove that the eigenvalues of the transpose $A^{\trans}$ are the same as the eigenvalues of $A$.

Let $A$ be an $n\times n$ invertible matrix. Then prove the transpose $A^{\trans}$ is also invertible and that the inverse matrix of the transpose $A^{\trans}$ is the transpose of the inverse matrix $A^{-1}$.

Namely, show that

\[(A^{\trans})^{-1}=(A^{-1})^{\trans}.\]

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.

Let $A$ be an $n\times n$ matrix. Suppose that $\mathbf{y}$ is a nonzero row vector such that

\[\mathbf{y}A=\mathbf{y}.\]
(Here a row vector means a $1\times n$ matrix.)

Prove that there is a nonzero column vector $\mathbf{x}$ such that

\[A\mathbf{x}=\mathbf{x}.\]
(Here a column vector means an $n \times 1$ matrix.)

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*)

Read solution

Suppose that $A$ is a real $n\times n$ matrix.

**(a)** Is it true that $A$ must commute with its transpose?

**(b)** Suppose that the columns of $A$ (considered as vectors) form an orthonormal set.

Is it true that the rows of $A$ must also form an orthonormal set?

(*University of California, Berkeley, Linear Algebra Qualifying Exam*)

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 & 1

\end{bmatrix}.\]
Then compute and simplify the following expression.

\[(A^{\trans}-B)^{\trans}+C(B^{-1}C)^{-1}.\]

(*The Ohio State University, Linear Algebra Midterm Exam Problem*)

Read solution

**(a)** The given matrix is the augmented matrix for a system of linear equations.

Give the vector form for the general solution.

\[ \left[\begin{array}{rrrrr|r}

1 & 0 & -1 & 0 &-2 & 0 \\

0 & 1 & 2 & 0 & -1 & 0 \\

0 & 0 & 0 & 1 & 1 & 0 \\

\end{array} \right].\]

**(b)** Let

\[A=\begin{bmatrix}

1 & 2 & 3 \\

4 &5 &6

\end{bmatrix}, B=\begin{bmatrix}

1 & 0 & 1 \\

0 &1 &0

\end{bmatrix}, C=\begin{bmatrix}

1 & 2\\

0& 6

\end{bmatrix}, \mathbf{v}=\begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}.\]
Then compute and simplify the following expression.

\[\mathbf{v}^{\trans}\left( A^{\trans}-(A-B)^{\trans}\right)C.\]

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 that this length is the distance between $\mathbf{a}$ and $\mathbf{b}$.)

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 orthogonal matrix.

**(c)** Find the eigenvalues of $A$.

Find the inverse matrix of the matrix

\[A=\begin{bmatrix}

\frac{2}{7} & \frac{3}{7} & \frac{6}{7} \\[6 pt]
\frac{6}{7} &\frac{2}{7} &-\frac{3}{7} \\[6pt]
-\frac{3}{7} & \frac{6}{7} & -\frac{2}{7}

\end{bmatrix}.\]

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.

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