# problems-in-mathematics-welcome-eye-catch

by Yu ·

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- Sum of Squares of Hermitian Matrices is Zero, then Hermitian Matrices Are All Zero Let $A_1, A_2, \dots, A_m$ be $n\times n$ Hermitian matrices. Show that if \[A_1^2+A_2^2+\cdots+A_m^2=\calO,\] where $\calO$ is the $n \times n$ zero matrix, then we have $A_i=\calO$ for each $i=1,2, \dots, m$. Hint. Recall that a complex matrix $A$ is Hermitian if […]
- 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 […]
- Symmetric Matrices and the Product of Two Matrices Let $A$ and $B$ be $n \times n$ real symmetric matrices. Prove the followings. (a) The product $AB$ is symmetric if and only if $AB=BA$. (b) If the product $AB$ is a diagonal matrix, then $AB=BA$. Hint. A matrix $A$ is called symmetric if $A=A^{\trans}$. In […]
- Every $n$-Dimensional Vector Space is Isomorphic to the Vector Space $\R^n$ Let $V$ be a vector space over the field of real numbers $\R$. Prove that if the dimension of $V$ is $n$, then $V$ is isomorphic to $\R^n$. Proof. Since $V$ is an $n$-dimensional vector space, it has a basis \[B=\{\mathbf{v}_1, \dots, […]
- Determine Dimensions of Eigenspaces From Characteristic Polynomial of Diagonalizable Matrix Let $A$ be an $n\times n$ matrix with the characteristic polynomial \[p(t)=t^3(t-1)^2(t-2)^5(t+2)^4.\] Assume that the matrix $A$ is diagonalizable. (a) Find the size of the matrix $A$. (b) Find the dimension of the eigenspace $E_2$ corresponding to the eigenvalue […]
- Determine All Matrices Satisfying Some Conditions on Eigenvalues and Eigenvectors Determine all $2\times 2$ matrices $A$ such that $A$ has eigenvalues $2$ and $-1$ with corresponding eigenvectors \[\begin{bmatrix} 1 \\ 0 \end{bmatrix} \text{ and } \begin{bmatrix} 2 \\ 1 \end{bmatrix},\] respectively. Solution. Suppose […]
- Every Group of Order 20449 is an Abelian Group Prove that every group of order $20449$ is an abelian group. Outline of the Proof Note that $20449=11^2 \cdot 13^2$. Let $G$ be a group of order $20449$. We prove by Sylow's theorem that there are a unique Sylow $11$-subgroup and a unique Sylow $13$-subgroup of […]
- Compute and Simplify the Matrix Expression Including Transpose and Inverse Matrices Let $A, B, C$ be the following $3\times 3$ matrices. \[A=\begin{bmatrix} 1 & 2 & 3 \\ 4 &5 &6 \\ 7 & 8 & 9 \end{bmatrix}, B=\begin{bmatrix} 1 & 0 & 1 \\ 0 &3 &0 \\ 1 & 0 & 5 \end{bmatrix}, C=\begin{bmatrix} -1 & 0\ & 1 \\ 0 &5 &6 \\ 3 & 0 & […]