## Idempotent (Projective) Matrices are Diagonalizable

## Problem 377

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

Then prove that $A$ is diagonalizable.

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

Then prove that $A$ is diagonalizable.

**(a)** Let

\[A=\begin{bmatrix}

0 & 0 & 0 & 0 \\

1 &1 & 1 & 1 \\

0 & 0 & 0 & 0 \\

1 & 1 & 1 & 1

\end{bmatrix}.\]
Find the eigenvalues of the matrix $A$. Also give the algebraic multiplicity of each eigenvalue.

**(b)** Let

\[A=\begin{bmatrix}

0 & 0 & 0 & 0 \\

1 &1 & 1 & 1 \\

0 & 0 & 0 & 0 \\

1 & 1 & 1 & 1

\end{bmatrix}.\]
One of the eigenvalues of the matrix $A$ is $\lambda=0$. Find the geometric multiplicity of the eigenvalue $\lambda=0$.

Let $n>1$ be a positive integer. Let $V=M_{n\times n}(\C)$ be the vector space over the complex numbers $\C$ consisting of all complex $n\times n$ matrices. The dimension of $V$ is $n^2$.

Let $A \in V$ and consider the set

\[S_A=\{I=A^0, A, A^2, \dots, A^{n^2-1}\}\]
of $n^2$ elements.

Prove that the set $S_A$ cannot be a basis of the vector space $V$ for any $A\in V$.

Let \[A=\begin{bmatrix}

a_0 & a_1 & \dots & a_{n-2} &a_{n-1} \\

a_{n-1} & a_0 & \dots & a_{n-3} & a_{n-2} \\

a_{n-2} & a_{n-1} & \dots & a_{n-4} & a_{n-3} \\

\vdots & \vdots & \dots & \vdots & \vdots \\

a_{2} & a_3 & \dots & a_{0} & a_{1}\\

a_{1} & a_2 & \dots & a_{n-1} & a_{0}

\end{bmatrix}\]
be a complex $n \times n$ matrix.

Such a matrix is called **circulant** matrix.

Then prove that the determinant of the circulant matrix $A$ is given by

\[\det(A)=\prod_{k=0}^{n-1}(a_0+a_1\zeta^k+a_2 \zeta^{2k}+\cdots+a_{n-1}\zeta^{k(n-1)}),\]
where $\zeta=e^{2 \pi i/n}$ is a primitive $n$-th root of unity.

Let $A$ be a $3\times 3$ matrix. Suppose that $A$ has eigenvalues $2$ and $-1$, and suppose that $\mathbf{u}$ and $\mathbf{v}$ are eigenvectors corresponding to $2$ and $-1$, respectively, where

\[\mathbf{u}=\begin{bmatrix}

1 \\

0 \\

-1

\end{bmatrix} \text{ and } \mathbf{v}=\begin{bmatrix}

2 \\

1 \\

0

\end{bmatrix}.\]
Then compute $A^5\mathbf{w}$, where

\[\mathbf{w}=\begin{bmatrix}

7 \\

2 \\

-3

\end{bmatrix}.\]

For each positive integer $n$, prove that the polynomial

\[(x-1)(x-2)\cdots (x-n)-1\]
is irreducible over the ring of integers $\Z$.

Let $S$ be the subset of $\R^4$ consisting of vectors $\begin{bmatrix}

x \\

y \\

z \\

w

\end{bmatrix}$ satisfying

\[2x+3y+5z+7w=0.\]
Then prove that the set $S$ is a subspace of $\R^4$.

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

Read solution

Let $T: \R^2 \to \R^2$ be a linear transformation such that

\[T\left(\, \begin{bmatrix}

1 \\

1

\end{bmatrix} \,\right)=\begin{bmatrix}

4 \\

1

\end{bmatrix}, T\left(\, \begin{bmatrix}

0 \\

1

\end{bmatrix} \,\right)=\begin{bmatrix}

3 \\

2

\end{bmatrix}.\]
Then find the matrix $A$ such that $T(\mathbf{x})=A\mathbf{x}$ for every $\mathbf{x}\in \R^2$, and find the rank and nullity of $T$.

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

Read solution

Let $T:\R^3 \to \R^2$ be a linear transformation such that

\[ T(\mathbf{e}_1)=\begin{bmatrix}

1 \\

0

\end{bmatrix}, T(\mathbf{e}_2)=\begin{bmatrix}

0 \\

1

\end{bmatrix}, T(\mathbf{e}_3)=\begin{bmatrix}

1 \\

0

\end{bmatrix},\]
where $\mathbf{e}_1, \mathbf{e}_2, \mathbf{e}_3$ are the standard basis of $\R^3$.

Then find the rank and the nullity of $T$.

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

Read solution

Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that

\[ T\left(\, \begin{bmatrix}

0 \\

1 \\

0

\end{bmatrix}\,\right) =\begin{bmatrix}

1 \\

2

\end{bmatrix} \text{ and }T\left(\, \begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix}\,\right)=\begin{bmatrix}

0 \\

1

\end{bmatrix}. \]
Then find $T\left(\, \begin{bmatrix}

0 \\

1 \\

2

\end{bmatrix} \,\right)$.

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

Read solution

Let $P_2$ be the vector space of all polynomials of degree $2$ or less with real coefficients.

Let

\[S=\{1+x+2x^2, \quad x+2x^2, \quad -1, \quad x^2\}\]
be the set of four vectors in $P_2$.

Then find a basis of the subspace $\Span(S)$ among the vectors in $S$.

(*Linear Algebra Exam Problem, the Ohio State University*)

Read solution

Let $A=\begin{bmatrix}

1 & 0 & 1 \\

0 &1 &0

\end{bmatrix}$.

**(a)** Find an orthonormal basis of the null space of $A$.

**(b)** Find the rank of $A$.

**(c)** Find an orthonormal basis of the row space of $A$.

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

Read solution

Let $f(x)=\sin^2(x)$, $g(x)=\cos^2(x)$, and $h(x)=1$. These are vectors in $C[-1, 1]$.

Determine whether the set $\{f(x), \, g(x), \, h(x)\}$ is linearly dependent or linearly independent.

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

Read solution

These are True or False problems.

For each of the following statements, determine if it contains a wrong information or not.

- Let $A$ be a $5\times 3$ matrix. Then the range of $A$ is a subspace in $\R^3$.
- The function $f(x)=x^2+1$ is not in the vector space $C[-1,1]$ because $f(0)=1\neq 0$.
- Since we have $\sin(x+y)=\sin(x)+\sin(y)$, the function $\sin(x)$ is a linear transformation.
- The set

\[\left\{\, \begin{bmatrix}

1 \\

0 \\

0

\end{bmatrix}, \begin{bmatrix}

0 \\

1 \\

1

\end{bmatrix} \,\right\}\] is an orthonormal set.

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

**(a)** Find all the eigenvalues and eigenvectors of the matrix

\[A=\begin{bmatrix}

3 & -2\\

6& -4

\end{bmatrix}.\]

**(b)** Let

\[A=\begin{bmatrix}

1 & 0 & 3 \\

4 &5 &6 \\

7 & 0 & 9

\end{bmatrix} \text{ and } B=\begin{bmatrix}

2 & 0 & 0 \\

0 & 3 &0 \\

0 & 0 & 4

\end{bmatrix}.\]
Then find the value of

\[\det(A^2B^{-1}A^{-2}B^2).\]
(For part (b) without computation, you may assume that $A$ and $B$ are invertible matrices.)

Let $n$ be an integer greater than $2$ and let $\zeta=e^{2\pi i/n}$ be a primitive $n$-th root of unity. Determine the degree of the extension of $\Q(\zeta)$ over $\Q(\zeta+\zeta^{-1})$.

The subfield $\Q(\zeta+\zeta^{-1})$ is called **maximal real subfield**.

Let

\[A=\begin{bmatrix}

3 & -12 & 4 \\

-1 &0 &-2 \\

-1 & 5 & -1

\end{bmatrix}.\]
Then find all eigenvalues of $A^5$. If $A$ is invertible, then find all the eigenvalues of $A^{-1}$.

Let $R$ be a commutative ring and let $I_1$ and $I_2$ be **comaximal ideals**. That is, we have

\[I_1+I_2=R.\]

Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal.

Add to solve laterLet $P$ be a $p$-group acting on a finite set $X$.

Let

\[ X^P=\{ x \in X \mid g\cdot x=x \text{ for all } g\in P \}. \]

The prove that

\[|X^P|\equiv |X| \pmod{p}.\]

Let $\alpha= \sqrt[3]{2}e^{2\pi i/3}$. Prove that $x_1^2+\cdots +x_k^2=-1$ has no solutions with all $x_i\in \Q(\alpha)$ and $k\geq 1$.

Add to solve later