# Diagonalization-eye-catch

• Matrix Operations with Transpose Calculate the following expressions, using the following matrices: $A = \begin{bmatrix} 2 & 3 \\ -5 & 1 \end{bmatrix}, \qquad B = \begin{bmatrix} 0 & -1 \\ 1 & -1 \end{bmatrix}, \qquad \mathbf{v} = \begin{bmatrix} 2 \\ -4 \end{bmatrix}$ (a) $A B^\trans + \mathbf{v} […] • Diagonalizable Matrix with Eigenvalue 1, -1 Suppose that$A$is a diagonalizable$n\times n$matrix and has only$1$and$-1$as eigenvalues. Show that$A^2=I_n$, where$I_n$is the$n\times n$identity matrix. (Stanford University Linear Algebra Exam) See below for a generalized problem. Hint. Diagonalize the […] • Galois Group of the Polynomial$x^2-2$Let$\Q$be the field of rational numbers. (a) Is the polynomial$f(x)=x^2-2$separable over$\Q$? (b) Find the Galois group of$f(x)$over$\Q$. Solution. (a) The polynomial$f(x)=x^2-2$is separable over$\Q$The roots of the polynomial$f(x)$are$\pm […]
• The Quadratic Integer Ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD) Prove that the quadratic integer ring $\Z[\sqrt{5}]$ is not a Unique Factorization Domain (UFD).   Proof. Every element of the ring $\Z[\sqrt{5}]$ can be written as $a+b\sqrt{5}$ for some integers $a, b$. The (field) norm $N$ of an element $a+b\sqrt{5}$ is […]
• Equivalent Conditions For a Prime Ideal in a Commutative Ring Let $R$ be a commutative ring and let $P$ be an ideal of $R$. Prove that the following statements are equivalent: (a) The ideal $P$ is a prime ideal. (b) For any two ideals $I$ and $J$, if $IJ \subset P$ then we have either $I \subset P$ or $J \subset P$.   Proof. […]
• A Rational Root of a Monic Polynomial with Integer Coefficients is an Integer Suppose that $\alpha$ is a rational root of a monic polynomial $f(x)$ in $\Z[x]$. Prove that $\alpha$ is an integer.   Proof. Suppose that $\alpha=\frac{p}{q}$ is a rational number in lowest terms, that is, $p$ and $q$ are relatively prime […]
• Group of Order $pq$ is Either Abelian or the Center is Trivial Let $G$ be a group of order $|G|=pq$, where $p$ and $q$ are (not necessarily distinct) prime numbers. Then show that $G$ is either abelian group or the center $Z(G)=1$. Hint. Use the result of the problem "If the Quotient by the Center is Cyclic, then the Group is […]
• Is a Set of All Nilpotent Matrix a Vector Space? Let $V$ denote the vector space of all real $n\times n$ matrices, where $n$ is a positive integer. Determine whether the set $U$ of all $n\times n$ nilpotent matrices is a subspace of the vector space $V$ or not.   Definition. An matrix $A$ is a nilpotent matrix if […]