Eigenvalues of a Matrix and Its Squared Matrix
Let $A$ be an $n \times n$ matrix. Suppose that the matrix $A^2$ has a real eigenvalue $\lambda>0$. Then show that either $\sqrt{\lambda}$ or $-\sqrt{\lambda}$ is an eigenvalue of the matrix $A$.
Hint.
Use the following fact: a scalar $\lambda$ is an eigenvalue of a […]
A Condition that a Vector is a Linear Combination of Columns Vectors of a Matrix
Suppose that an $n \times m$ matrix $M$ is composed of the column vectors $\mathbf{b}_1 , \cdots , \mathbf{b}_m$.
Prove that a vector $\mathbf{v} \in \R^n$ can be written as a linear combination of the column vectors if and only if there is a vector $\mathbf{x}$ which solves the […]
Prove a Group is Abelian if $(ab)^3=a^3b^3$ and No Elements of Order $3$
Let $G$ be a group. Suppose that we have
\[(ab)^3=a^3b^3\]
for any elements $a, b$ in $G$. Also suppose that $G$ has no elements of order $3$.
Then prove that $G$ is an abelian group.
Proof.
Let $a, b$ be arbitrary elements of the group $G$. We want […]
A Ring Has Infinitely Many Nilpotent Elements if $ab=1$ and $ba \neq 1$
Let $R$ be a ring with $1$.
Suppose that $a, b$ are elements in $R$ such that
\[ab=1 \text{ and } ba\neq 1.\]
(a) Prove that $1-ba$ is idempotent.
(b) Prove that $b^n(1-ba)$ is nilpotent for each positive integer $n$.
(c) Prove that the ring $R$ has infinitely many […]
A Symmetric Positive Definite Matrix and An Inner Product on a Vector Space
(a) Suppose that $A$ is an $n\times n$ real symmetric positive definite matrix.
Prove that
\[\langle \mathbf{x}, \mathbf{y}\rangle:=\mathbf{x}^{\trans}A\mathbf{y}\]
defines an inner product on the vector space $\R^n$.
(b) Let $A$ be an $n\times n$ real matrix. Suppose […]