# linear-algebra-eye-catch3

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- Prove a Group is Abelian if $(ab)^2=a^2b^2$ Let $G$ be a group. Suppose that \[(ab)^2=a^2b^2\] for any elements $a, b$ in $G$. Prove that $G$ is an abelian group. Proof. To prove that $G$ is an abelian group, we need \[ab=ba\] for any elements $a, b$ in $G$. By the given […]
- Inequality about Eigenvalue of a Real Symmetric Matrix Let $A$ be an $n\times n$ real symmetric matrix. Prove that there exists an eigenvalue $\lambda$ of $A$ such that for any vector $\mathbf{v}\in \R^n$, we have the inequality \[\mathbf{v}\cdot A\mathbf{v} \leq \lambda \|\mathbf{v}\|^2.\] Proof. Recall […]
- Solve a System of Linear Equations by Gauss-Jordan Elimination Solve the following system of linear equations using Gauss-Jordan elimination. \begin{align*} 6x+8y+6z+3w &=-3 \\ 6x-8y+6z-3w &=3\\ 8y \,\,\,\,\,\,\,\,\,\,\,- 6w &=6 \end{align*} We use the following notation. Elementary row operations. The […]
- Conjugate of the Centralizer of a Set is the Centralizer of the Conjugate of the Set Let $X$ be a subset of a group $G$. Let $C_G(X)$ be the centralizer subgroup of $X$ in $G$. For any $g \in G$, show that $gC_G(X)g^{-1}=C_G(gXg^{-1})$. Proof. $(\subset)$ We first show that $gC_G(X)g^{-1} \subset C_G(gXg^{-1})$. Take any $h\in C_G(X)$. Then for […]
- Generators of the Augmentation Ideal in a Group Ring Let $R$ be a commutative ring with $1$ and let $G$ be a finite group with identity element $e$. Let $RG$ be the group ring. Then the map $\epsilon: RG \to R$ defined by \[\epsilon(\sum_{i=1}^na_i g_i)=\sum_{i=1}^na_i,\] where $a_i\in R$ and $G=\{g_i\}_{i=1}^n$, is a ring […]
- Condition that a Matrix is Similar to the Companion Matrix of its Characteristic Polynomial Let $A$ be an $n\times n$ complex matrix. Let $p(x)=\det(xI-A)$ be the characteristic polynomial of $A$ and write it as \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Let $C$ be the companion matrix of the polynomial $p(x)$ given […]
- The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$ Let $F$ be a field and let \[H(F)=\left\{\, \begin{bmatrix} 1 & a & b \\ 0 &1 &c \\ 0 & 0 & 1 \end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}\] be the Heisenberg group over $F$. (The group operation of the Heisenberg group is matrix […]
- Inverse Matrix Contains Only Integers if and only if the Determinant is $\pm 1$ Let $A$ be an $n\times n$ nonsingular matrix with integer entries. Prove that the inverse matrix $A^{-1}$ contains only integer entries if and only if $\det(A)=\pm 1$. Hint. If $B$ is a square matrix whose entries are integers, then the […]