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  • Eigenvalues and Algebraic/Geometric Multiplicities of Matrix $A+cI$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 IndexSubgroup 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 $G=AB$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 SystemPossibilities 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 $A^3=O$.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 FieldPolynomial $x^p-x+a$ 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 $\mathbf{v}$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 TransformationsEigenvalues 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 […]

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