# graphs-of-characteristic-polynomials

### More from my site

• A Line is a Subspace if and only if its $y$-Intercept is Zero Let $\R^2$ be the $x$-$y$-plane. Then $\R^2$ is a vector space. A line $\ell \subset \mathbb{R}^2$ with slope $m$ and $y$-intercept $b$ is defined by $\ell = \{ (x, y) \in \mathbb{R}^2 \mid y = mx + b \} .$ Prove that $\ell$ is a subspace of $\mathbb{R}^2$ if and only if $b = […] • Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible. Let $A=\begin{bmatrix} 1 & 3 & 3 \\ -3 &-5 &-3 \\ 3 & 3 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 4 & 3 \\ -4 &-6 &-3 \\ 3 & 3 & 1 \end{bmatrix}.$ For this problem, you may use the fact that both matrices have the same characteristic […] • Group of Invertible Matrices Over a Finite Field and its Stabilizer Let$\F_p$be the finite field of$p$elements, where$p$is a prime number. Let$G_n=\GL_n(\F_p)$be the group of$n\times n$invertible matrices with entries in the field$\F_p$. As usual in linear algebra, we may regard the elements of$G_n$as linear transformations on$\F_p^n$, […] • If the Matrix Product$AB=0$, then is$BA=0$as Well? Let$A$and$B$be$n\times n$matrices. Suppose that the matrix product$AB=O$, where$O$is the$n\times n$zero matrix. Is it true that the matrix product with opposite order$BA$is also the zero matrix? If so, give a proof. If not, give a […] • Is the Product of a Nilpotent Matrix and an Invertible Matrix Nilpotent? A square matrix$A$is called nilpotent if there exists a positive integer$k$such that$A^k=O$, where$O$is the zero matrix. (a) If$A$is a nilpotent$n \times n$matrix and$B$is an$n\times n$matrix such that$AB=BA$. Show that the product$AB$is nilpotent. (b) Let$P$[…] • Transpose of a Matrix and Eigenvalues and Related Questions Let$A$be an$n \times n$real matrix. Prove the followings. (a) The matrix$AA^{\trans}$is a symmetric matrix. (b) The set of eigenvalues of$A$and the set of eigenvalues of$A^{\trans}$are equal. (c) The matrix$AA^{\trans}$is non-negative definite. (An$n\times n$[…] • Every Maximal Ideal of a Commutative Ring is a Prime Ideal Let$R$be a commutative ring with unity. Then show that every maximal ideal of$R$is a prime ideal. We give two proofs. Proof 1. The first proof uses the following facts. Fact 1. An ideal$I$of$R$is a prime ideal if and only if$R/I$is an integral […] • If a Subgroup Contains a Sylow Subgroup, then the Normalizer is the Subgroup itself Let$G$be a finite group and$P$be a nontrivial Sylow subgroup of$G$. Let$H$be a subgroup of$G$containing the normalizer$N_G(P)$of$P$in$G$. Then show that$N_G(H)=H\$.   Hint. Use the conjugate part of the Sylow theorem. See the second statement of the […]