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  • Maximize the Dimension of the Null Space of $A-aI$Maximize the Dimension of the Null Space of $A-aI$ Let \[ A=\begin{bmatrix} 5 & 2 & -1 \\ 2 &2 &2 \\ -1 & 2 & 5 \end{bmatrix}.\] Pick your favorite number $a$. Find the dimension of the null space of the matrix $A-aI$, where $I$ is the $3\times 3$ identity matrix. Your score of this problem is equal to that […]
  • Idempotent (Projective) Matrices are DiagonalizableIdempotent (Projective) Matrices are Diagonalizable Let $A$ be an $n\times n$ idempotent complex matrix. Then prove that $A$ is diagonalizable.   Definition. An $n\times n$ matrix $A$ is said to be idempotent if $A^2=A$. It is also called projective matrix. Proof. In general, an $n \times n$ matrix $B$ is […]
  • The Subspace of Linear Combinations whose Sums of Coefficients are zeroThe Subspace of Linear Combinations whose Sums of Coefficients are zero Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset \[W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […]
  • The Order of a Conjugacy Class Divides the Order of the GroupThe 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 $a$ […]
  • Matrices Satisfying the Relation $HE-EH=2E$Matrices Satisfying the Relation $HE-EH=2E$ Let $H$ and $E$ be $n \times n$ matrices satisfying the relation \[HE-EH=2E.\] Let $\lambda$ be an eigenvalue of the matrix $H$ such that the real part of $\lambda$ is the largest among the eigenvalues of $H$. Let $\mathbf{x}$ be an eigenvector corresponding to $\lambda$. Then […]
  • Finite Integral Domain is a FieldFinite Integral Domain is a Field Show that any finite integral domain $R$ is a field.   Definition. A commutative ring $R$ with $1\neq 0$ is called an integral domain if it has no zero divisors. That is, if $ab=0$ for $a, b \in R$, then either $a=0$ or $b=0$. Proof. We give two proofs. Proof […]
  • Linearly Independent/Dependent Vectors QuestionLinearly Independent/Dependent Vectors Question Let $V$ be an $n$-dimensional vector space over a field $K$. Suppose that $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ are linearly independent vectors in $V$. Are the following vectors linearly independent? \[\mathbf{v}_1+\mathbf{v}_2, \quad \mathbf{v}_2+\mathbf{v}_3, […]
  • Eigenvalues of Squared Matrix and Upper Triangular MatrixEigenvalues of Squared Matrix and Upper Triangular Matrix Suppose that $A$ and $P$ are $3 \times 3$ matrices and $P$ is invertible matrix. If \[P^{-1}AP=\begin{bmatrix} 1 & 2 & 3 \\ 0 &4 &5 \\ 0 & 0 & 6 \end{bmatrix},\] then find all the eigenvalues of the matrix $A^2$.   We give two proofs. The first version is a […]

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