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  • Companion Matrix for a PolynomialCompanion Matrix for a Polynomial Consider a polynomial \[p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,\] where $a_i$ are real numbers. Define the matrix \[A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […]
  • Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$Find the Nullity of the Matrix $A+I$ if Eigenvalues are $1, 2, 3, 4, 5$ Let $A$ be an $n\times n$ matrix. Its only eigenvalues are $1, 2, 3, 4, 5$, possibly with multiplicities. What is the nullity of the matrix $A+I_n$, where $I_n$ is the $n\times n$ identity matrix? (The Ohio State University, Linear Algebra Final Exam […]
  • Find the Vector Form Solution to the Matrix Equation $A\mathbf{x}=\mathbf{0}$Find the Vector Form Solution to the Matrix Equation $A\mathbf{x}=\mathbf{0}$ Find the vector form solution $\mathbf{x}$ of the equation $A\mathbf{x}=\mathbf{0}$, where $A=\begin{bmatrix} 1 & 1 & 1 & 1 &2 \\ 1 & 2 & 4 & 0 & 5 \\ 3 & 2 & 0 & 5 & 2 \\ \end{bmatrix}$. Also, find two linearly independent vectors $\mathbf{x}$ satisfying […]
  • A Symmetric Positive Definite Matrix and An Inner Product on a Vector SpaceA 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 […]
  • If a Subgroup $H$ is in the Center of a Group $G$ and $G/H$ is Nilpotent, then $G$ is NilpotentIf a Subgroup $H$ is in the Center of a Group $G$ and $G/H$ is Nilpotent, then $G$ is Nilpotent Let $G$ be a nilpotent group and let $H$ be a subgroup such that $H$ is a subgroup in the center $Z(G)$ of $G$. Suppose that the quotient $G/H$ is nilpotent. Then show that $G$ is also nilpotent.   Definition (Nilpotent Group) We recall here the definition of a […]
  • There is at Least One Real Eigenvalue of an Odd Real MatrixThere is at Least One Real Eigenvalue of an Odd Real Matrix Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix. Prove that the matrix $A$ has at least one real eigenvalue.   We give two proofs. Proof 1. Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$. It is a degree $n$ […]
  • Mathematics About the Number 2018Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018.   Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is \[2018=2\cdot 1009.\] Here $2$ and $1009$ are […]
  • Subgroup Containing All $p$-Sylow Subgroups of a GroupSubgroup Containing All $p$-Sylow Subgroups of a Group Suppose that $G$ is a finite group of order $p^an$, where $p$ is a prime number and $p$ does not divide $n$. Let $N$ be a normal subgroup of $G$ such that the index $|G: N|$ is relatively prime to $p$. Then show that $N$ contains all $p$-Sylow subgroups of […]

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