If the Kernel of a Matrix $A$ is Trivial, then $A^T A$ is Invertible

Problem 38

Let $A$ be an $m \times n$ real matrix.
Then the kernel of $A$ is defined as $\ker(A)=\{ x\in \R^n \mid Ax=0 \}$.

The kernel is also called the null space of $A$.
Suppose that $A$ is an $m \times n$ real matrix such that $\ker(A)=0$. Prove that $A^{\trans}A$ is invertible.

(Stanford University Linear Algebra Exam)

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Find a Formula for a Linear Transformation

Problem 36

If $L:\R^2 \to \R^3$ is a linear transformation such that
\begin{align*}
L\left( \begin{bmatrix}
1 \\
0
\end{bmatrix}\right)
=\begin{bmatrix}
1 \\
1 \\
2
\end{bmatrix}, \,\,\,\,
L\left( \begin{bmatrix}
1 \\
1
\end{bmatrix}\right)
=\begin{bmatrix}
2 \\
3 \\
2
\end{bmatrix}.
\end{align*}
then

(a) find $L\left( \begin{bmatrix}
1 \\
2
\end{bmatrix}\right)$, and

(b) find the formula for $L\left( \begin{bmatrix}
x \\
y
\end{bmatrix}\right)$.

 

If you think you can solve (b), then skip (a) and solve (b) first and use the result of (b) to answer (a).

(Part (a) is an exam problem of Purdue University)

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Find the Rank of the Matrix $A+I$ if Eigenvalues of $A$ are $1, 2, 3, 4, 5$

Problem 35

Let $A$ be an $n$ by $n$ matrix with entries in complex numbers $\C$. Its only eigenvalues are $1,2,3,4,5$, possibly with multiplicities. What is the rank of the matrix $A+I_n$, where $I_n$ is the identity $n$ by $n$ matrix.

(UCB-University of California, Berkeley, Exam)

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Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors

Problem 34

(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.

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The Subspace of Matrices that are Diagonalized by a Fixed Matrix

Problem 33

Suppose that $S$ is a fixed invertible $3$ by $3$ matrix. This question is about all the matrices $A$ that are diagonalized by $S$, so that $S^{-1}AS$ is diagonal. Show that these matrices $A$ form a subspace of $3$ by $3$ matrix space.

(MIT-Massachusetts Institute of Technology Exam)

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Equivalent Conditions to be a Unitary Matrix

Problem 29

A complex matrix is called unitary if $\overline{A}^{\trans} A=I$.

The inner product $(\mathbf{x}, \mathbf{y})$ of complex vector $\mathbf{x}$, $\mathbf{y}$ is defined by $(\mathbf{x}, \mathbf{y}):=\overline{\mathbf{x}}^{\trans} \mathbf{y}$. The length of a complex vector $\mathbf{x}$ is defined to be $||\mathbf{x}||:=\sqrt{(\mathbf{x}, \mathbf{x})}$.

Let $A$ be an $n \times n$ complex matrix. Prove that the followings are equivalent.

(a) The matrix $A$ is unitary.

(b) $||A \mathbf{x}||=|| \mathbf{x}||$ for any $n$-dimensional complex vector $\mathbf{x}$.

(c) $(A\mathbf{x}, A\mathbf{y})=(\mathbf{x}, \mathbf{y})$ for any $n$-dimensional complex vectors $x, y$

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Finite Order Matrix and its Trace

Problem 28

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 $A=I_n$.

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A Matrix is Invertible If and Only If It is Nonsingular

Problem 26

In this problem, we will show that the concept of non-singularity of a matrix is equivalent to the concept of invertibility.
That is, we will prove that:

A matrix $A$ is nonsingular if and only if $A$ is invertible.

(a) Show that if $A$ is invertible, then $A$ is nonsingular.


(b) Let $A, B, C$ be $n\times n$ matrices such that $AB=C$.
Prove that if either $A$ or $B$ is singular, then so is $C$.


(c) Show that if $A$ is nonsingular, then $A$ is invertible.

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Properties of Nonsingular and Singular Matrices

Problem 25

An $n \times n$ matrix $A$ is called nonsingular if the only solution of the equation $A \mathbf{x}=\mathbf{0}$ is the zero vector $\mathbf{x}=\mathbf{0}$.
Otherwise $A$ is called singular.

(a) Show that if $A$ and $B$ are $n\times n$ nonsingular matrices, then the product $AB$ is also nonsingular.

(b) Show that if $A$ is nonsingular, then the column vectors of $A$ are linearly independent.

(c) Show that an $n \times n$ matrix $A$ is nonsingular if and only if the equation $A\mathbf{x}=\mathbf{b}$ has a unique solution for any vector $\mathbf{b}\in \R^n$.

Restriction
Do not use the fact that a matrix is nonsingular if and only if the matrix is invertible.

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Basic Properties of Characteristic Groups

Problem 22

Definition (automorphism).

An isomorphism from a group $G$ to itself is called an automorphism of $G$.
The set of all automorphism is denoted by $\Aut(G)$.

Definition (characteristic subgroup).

A subgroup $H$ of a group $G$ is called characteristic in $G$ if for any $\phi \in \Aut(G)$, we have $\phi(H)=H$. In words, this means that each automorphism of $G$ maps $H$ to itself.

Prove the followings.

(a) If $H$ is characteristic in $G$, then $H$ is a normal subgroup of $G$.

(b) If $H$ is the unique subgroup of $G$ of a given order, then $H$ is characteristic in $G$.

(c) Suppose that a subgroup $K$ is characteristic in a group $H$ and $H$ is a normal subgroup of $G$. Then $K$ is a normal subgroup in $G$.

 
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A Group of Order the Square of a Prime is Abelian

Problem 20

Suppose the order of a group $G$ is $p^2$, where $p$ is a prime number.
Show that

(a) the group $G$ is an abelian group, and

(b) the group $G$ is isomorphic to either $\Zmod{p^2}$ or $\Zmod{p} \times \Zmod{p}$ without using the fundamental theorem of abelian groups.

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Questions About the Trace of a Matrix

Problem 19

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 $\tr(A^2)>0$.

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