graphs-of-characteristic-polynomials

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Graphs of characteristic polynomials


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  • Diagonalize the 3 by 3 Matrix if it is DiagonalizableDiagonalize the 3 by 3 Matrix if it is Diagonalizable Determine whether the matrix \[A=\begin{bmatrix} 0 & 1 & 0 \\ -1 &0 &0 \\ 0 & 0 & 2 \end{bmatrix}\] is diagonalizable. If it is diagonalizable, then find the invertible matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$.   How to […]
  • How to Prove a Matrix is Nonsingular in 10 SecondsHow to Prove a Matrix is Nonsingular in 10 Seconds Using the numbers appearing in \[\pi=3.1415926535897932384626433832795028841971693993751058209749\dots\] we construct the matrix \[A=\begin{bmatrix} 3 & 14 &1592& 65358\\ 97932& 38462643& 38& 32\\ 7950& 2& 8841& 9716\\ 939937510& 5820& 974& […]
  • The Rank of the Sum of Two MatricesThe Rank of the Sum of Two Matrices Let $A$ and $B$ be $m\times n$ matrices. Prove that \[\rk(A+B) \leq \rk(A)+\rk(B).\] Proof. Let \[A=[\mathbf{a}_1, \dots, \mathbf{a}_n] \text{ and } B=[\mathbf{b}_1, \dots, \mathbf{b}_n],\] where $\mathbf{a}_i$ and $\mathbf{b}_i$ are column vectors of $A$ and $B$, […]
  • Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$Every Sylow 11-Subgroup of a Group of Order 231 is Contained in the Center $Z(G)$ Let $G$ be a finite group of order $231=3\cdot 7 \cdot 11$. Prove that every Sylow $11$-subgroup of $G$ is contained in the center $Z(G)$. Hint. Prove that there is a unique Sylow $11$-subgroup of $G$, and consider the action of $G$ on the Sylow $11$-subgroup by […]
  • A Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero EigenvaluesA Linear Transformation Preserves Exactly Two Lines If and Only If There are Two Real Non-Zero Eigenvalues Let $T:\R^2 \to \R^2$ be a linear transformation and let $A$ be the matrix representation of $T$ with respect to the standard basis of $\R^2$. Prove that the following two statements are equivalent. (a) There are exactly two distinct lines $L_1, L_2$ in $\R^2$ passing through […]
  • True or False: Eigenvalues of a Real Matrix Are Real NumbersTrue or False: Eigenvalues of a Real Matrix Are Real Numbers Answer the following questions regarding eigenvalues of a real matrix. (a) True or False. If each entry of an $n \times n$ matrix $A$ is a real number, then the eigenvalues of $A$ are all real numbers. (b) Find the eigenvalues of the matrix \[B=\begin{bmatrix} -2 & […]
  • Determine a Value of Linear Transformation From $\R^3$ to $\R^2$Determine a Value of Linear Transformation From $\R^3$ to $\R^2$ Let $T$ be a linear transformation from $\R^3$ to $\R^2$ such that \[ T\left(\, \begin{bmatrix} 0 \\ 1 \\ 0 \end{bmatrix}\,\right) =\begin{bmatrix} 1 \\ 2 \end{bmatrix} \text{ and }T\left(\, \begin{bmatrix} 0 \\ 1 \\ 1 […]
  • Vector Space of 2 by 2 Traceless MatricesVector Space of 2 by 2 Traceless Matrices Let $V$ be the vector space of all $2\times 2$ matrices whose entries are real numbers. Let \[W=\left\{\, A\in V \quad \middle | \quad A=\begin{bmatrix} a & b\\ c& -a \end{bmatrix} \text{ for any } a, b, c\in \R \,\right\}.\] (a) Show that $W$ is a subspace of […]

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