Every Diagonalizable Nilpotent Matrix is the Zero Matrix
Prove that if $A$ is a diagonalizable nilpotent matrix, then $A$ is the zero matrix $O$.
Definition (Nilpotent Matrix)
A square matrix $A$ is called nilpotent if there exists a positive integer $k$ such that $A^k=O$.
Proof.
Main Part
Since $A$ is […]
A Matrix Equation of a Symmetric Matrix and the Limit of its Solution
Let $A$ be a real symmetric $n\times n$ matrix with $0$ as a simple eigenvalue (that is, the algebraic multiplicity of the eigenvalue $0$ is $1$), and let us fix a vector $\mathbf{v}\in \R^n$.
(a) Prove that for sufficiently small positive real $\epsilon$, the equation […]
Construction of a Symmetric Matrix whose Inverse Matrix is Itself
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 […]
Normal Subgroups Intersecting Trivially Commute in a Group
Let $A$ and $B$ be normal subgroups of a group $G$. Suppose $A\cap B=\{e\}$, where $e$ is the unit element of the group $G$.
Show that for any $a \in A$ and $b \in B$ we have $ab=ba$.
Hint.
Consider the commutator of $a$ and $b$, that […]
The Preimage of Prime ideals are Prime Ideals
Let $f: R\to R'$ be a ring homomorphism. Let $P$ be a prime ideal of the ring $R'$.
Prove that the preimage $f^{-1}(P)$ is a prime ideal of $R$.
Proof.
The preimage of an ideal by a ring homomorphism is an ideal.
(See the post "The inverse image of an ideal by […]
The Order of a Conjugacy Class Divides the Order of the Group
Let $G$ be a finite group.
The centralizer of an element $a$ of $G$ is defined to be
\[C_G(a)=\{g\in G \mid ga=ag\}.\]
A conjugacy class is a set of the form
\[\Cl(a)=\{bab^{-1} \mid b\in G\}\]
for some $a\in G$.
(a) Prove that the centralizer of an element of $a$ […]
Isomorphism Criterion of Semidirect Product of Groups
Let $A$, $B$ be groups. Let $\phi:B \to \Aut(A)$ be a group homomorphism.
The semidirect product $A \rtimes_{\phi} B$ with respect to $\phi$ is a group whose underlying set is $A \times B$ with group operation
\[(a_1, b_1)\cdot (a_2, b_2)=(a_1\phi(b_1)(a_2), b_1b_2),\]
where $a_i […]