## 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 $A$ and $B$ be $n \times n$ matrices.

Is it always true that $\tr (A B) = \tr (A) \tr (B) $?

If it is true, prove it. If not, give a counterexample.

Add to solve laterLet $A$ 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=\begin{bmatrix}

a & b\\

c& d

\end{bmatrix}$ be an $2\times 2$ matrix.

Express the eigenvalues of $A$ in terms of the trace and the determinant of $A$.

Add to solve later Let $I$ be the $2\times 2$ identity matrix.

Then prove that $-I$ cannot be a commutator $[A, B]:=ABA^{-1}B^{-1}$ for any $2\times 2$ matrices $A$ and $B$ with determinant $1$.

Let $A$ and $B$ be $2\times 2$ matrices such that $(AB)^2=O$, where $O$ is the $2\times 2$ zero matrix.

Determine whether $(BA)^2$ must be $O$ as well. If so, prove it. If not, give a counter example.

Add to solve later Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix.

Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula:

\[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\]

Using the formula, calculate the inverse matrix of $\begin{bmatrix}

2 & 1\\

1& 2

\end{bmatrix}$.

**(a)** Is the matrix $A=\begin{bmatrix}

1 & 2\\

0& 3

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

3 & 0\\

1& 2

\end{bmatrix}$?

**(b)** Is the matrix $A=\begin{bmatrix}

0 & 1\\

5& 3

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

1 & 2\\

4& 3

\end{bmatrix}$?

**(c)** Is the matrix $A=\begin{bmatrix}

-1 & 6\\

-2& 6

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

3 & 0\\

0& 2

\end{bmatrix}$?

**(d)** Is the matrix $A=\begin{bmatrix}

-1 & 6\\

-2& 6

\end{bmatrix}$ similar to the matrix $B=\begin{bmatrix}

1 & 2\\

-1& 4

\end{bmatrix}$?

Prove that if $A$ and $B$ are similar matrices, then their determinants are the same.

Add to solve later**(a)** A $2 \times 2$ matrix $A$ satisfies $\tr(A^2)=5$ and $\tr(A)=3$.

Find $\det(A)$.

**(b)** A $2 \times 2$ matrix has two parallel columns and $\tr(A)=5$. Find $\tr(A^2)$.

**(c)** A $2\times 2$ matrix $A$ has $\det(A)=5$ and positive integer eigenvalues. What is the trace of $A$?

(*Harvard University, Linear Algebra Exam Problem*)

Let $A, B$ be complex $2\times 2$ matrices satisfying the relation

\[A=AB-BA.\]

Prove that $A^2=O$, where $O$ is the $2\times 2$ zero matrix.

Add to solve laterLet $I$ be the $n\times n$ identity matrix, where $n$ is a positive integer. Prove that there are no $n\times n$ matrices $X$ and $Y$ such that

\[XY-YX=I.\]

Read solution

Let $V$ be the set of all $n \times n$ diagonal matrices whose traces are zero.

That is,

\begin{equation*}

V:=\left\{ A=\begin{bmatrix}

a_{11} & 0 & \dots & 0 \\

0 &a_{22} & \dots & 0 \\

0 & 0 & \ddots & \vdots \\

0 & 0 & \dots & a_{nn}

\end{bmatrix} \quad \middle| \quad

\begin{array}{l}

a_{11}, \dots, a_{nn} \in \C,\\

\tr(A)=0 \\

\end{array}

\right\}

\end{equation*}

Let $E_{ij}$ denote the $n \times n$ matrix whose $(i,j)$-entry is $1$ and zero elsewhere.

**(a)** Show that $V$ is a subspace of the vector space $M_n$ over $\C$ of all $n\times n$ matrices. (You may assume without a proof that $M_n$ is a vector space.)

**(b)** Show that matrices

\[E_{11}-E_{22}, \, E_{22}-E_{33}, \, \dots,\, E_{n-1\, n-1}-E_{nn}\]
are a basis for the vector space $V$.

**(c)** Find the dimension of $V$.

Read solution

Let $F$ and $H$ be an $n\times n$ matrices satisfying the relation

\[HF-FH=-2F.\]

**(a)** Find the trace of the matrix $F$.

**(b)** Let $\lambda$ be an eigenvalue of $H$ and let $\mathbf{v}$ be an eigenvector corresponding to $\lambda$. Show that there exists an positive integer $N$ such that $F^N\mathbf{v}=\mathbf{0}$.

Let $A$ be an $n\times n$ matrix such that $A^k=I_n$, where $k\in \N$ and $I_n$ is the $n \times n$ identity matrix.

Show that the trace of $(A^{-1})^{\trans}$ is the conjugate of the trace of $A$. That is, show that $\tr((A^{-1})^{\trans})=\overline{\tr(A)}$.

Add to solve later

**(a)** Let

\[A=\begin{bmatrix}

a_{11} & a_{12}\\

a_{21}& a_{22}

\end{bmatrix}\]
be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$.

(Such a matrix is called (right) * stochastic matrix* (also termed probability matrix, transition matrix, substitution matrix, or Markov matrix).)

Then prove that the matrix $A$ has an eigenvalue $1$.

**(b)** Find all the eigenvalues of the matrix

\[B=\begin{bmatrix}

0.3 & 0.7\\

0.6& 0.4

\end{bmatrix}.\]

**(c)** For each eigenvalue of $B$, find the corresponding eigenvectors.

Let $A$ be an $n\times n$ matrix and suppose that $A^r=I_n$ for some positive integer $r$. Then show that

**(a)** $|\tr(A)|\leq n$.

**(b)** If $|\tr(A)|=n$, then $A=\zeta I_n$ for an $r$-th root of unity $\zeta$.

**(c)** $\tr(A)=n$ if and only if $A=I_n$.

Let $A$ be an $n \times n$ matrix such that $\tr(A^n)=0$ for all $n \in \N$.

Then prove that $A$ is a nilpotent matrix. Namely there exist a positive integer $m$ such that $A^m$ is the zero matrix.

Let $A=(a_{i j})$ and $B=(b_{i j})$ be $n\times n$ real matrices for some $n \in \N$. Then answer the following questions about the trace of a matrix.

**(a)** Express $\tr(AB^{\trans})$ in terms of the entries of the matrices $A$ and $B$. Here $B^{\trans}$ is the transpose matrix of $B$.

**(b)** Show that $\tr(AA^{\trans})$ is the sum of the square of the entries of $A$.

**(c)** Show that if $A$ is nonzero symmetric matrix, then $\tr(A^2)>0$.