# symmetric-matrix

### More from my site

• Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$ Let $A$ be an $n \times n$ matrix and let $c$ be a complex number. (a) For each eigenvalue $\lambda$ of $A$, prove that $\lambda+c$ is an eigenvalue of the matrix $A+cI$, where $I$ is the identity matrix. What can you say about the eigenvectors corresponding to […]
• Subgroup of Finite Index Contains a Normal Subgroup of Finite Index Let $G$ be a group and let $H$ be a subgroup of finite index. Then show that there exists a normal subgroup $N$ of $G$ such that $N$ is of finite index in $G$ and $N\subset H$.   Proof. The group $G$ acts on the set of left cosets $G/H$ by left multiplication. Hence […]
• If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$ Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying $|A|+|B| > |G|.$ Here $|X|$ denotes the cardinality (the number of elements) of the set $X$. Then prove that $G=AB$, where $AB=\{ab \mid a\in A, b\in B\}.$   Proof. Since $A, B$ […]
• Possibilities For the Number of Solutions for a Linear System Determine whether the following systems of equations (or matrix equations) described below has no solution, one unique solution or infinitely many solutions and justify your answer. (a) $\left\{ \begin{array}{c} ax+by=c \\ dx+ey=f, \end{array} \right.$ where $a,b,c, d$ […]
• Example of a Nilpotent Matrix $A$ such that $A^2\neq O$ but $A^3=O$. Find a nonzero $3\times 3$ matrix $A$ such that $A^2\neq O$ and $A^3=O$, where $O$ is the $3\times 3$ zero matrix. (Such a matrix is an example of a nilpotent matrix. See the comment after the solution.)   Solution. For example, let $A$ be the following $3\times […] • Polynomial$x^p-x+a$is Irreducible and Separable Over a Finite Field Let$p\in \Z$be a prime number and let$\F_p$be the field of$p$elements. For any nonzero element$a\in \F_p$, prove that the polynomial $f(x)=x^p-x+a$ is irreducible and separable over$F_p$. (Dummit and Foote "Abstract Algebra" Section 13.5 Exercise #5 on […] • Prove that$\mathbf{v} \mathbf{v}^\trans$is a Symmetric Matrix for any Vector$\mathbf{v}$Let$\mathbf{v}$be an$n \times 1$column vector. Prove that$\mathbf{v} \mathbf{v}^\trans$is a symmetric matrix. Definition (Symmetric Matrix). A matrix$A$is called symmetric if$A^{\trans}=A$. In terms of entries, an$n\times n$matrix$A=(a_{ij})$is […] • Eigenvalues of Similarity Transformations Let$A$be an$n\times n$complex matrix. Let$S$be an invertible matrix. (a) If$SAS^{-1}=\lambda A$for some complex number$\lambda$, then prove that either$\lambda^n=1$or$A$is a singular matrix. (b) If$n$is odd and$SAS^{-1}=-A$, then prove that$0\$ is an […]