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- The Set of Square Elements in the Multiplicative Group $(\Zmod{p})^*$ Suppose that $p$ is a prime number greater than $3$. Consider the multiplicative group $G=(\Zmod{p})^*$ of order $p-1$. (a) Prove that the set of squares $S=\{x^2\mid x\in G\}$ is a subgroup of the multiplicative group $G$. (b) Determine the index $[G : S]$. (c) Assume […]
- A Positive Definite Matrix Has a Unique Positive Definite Square Root Prove that a positive definite matrix has a unique positive definite square root. In this post, we review several definitions (a square root of a matrix, a positive definite matrix) and solve the above problem. After the proof, several extra problems about square […]
- Every Finite Group Having More than Two Elements Has a Nontrivial Automorphism Prove that every finite group having more than two elements has a nontrivial automorphism. (Michigan State University, Abstract Algebra Qualifying Exam) Proof. Let $G$ be a finite group and $|G|> 2$. Case When $G$ is a Non-Abelian Group Let us first […]
- The Inner Product on $\R^2$ induced by a Positive Definite Matrix and Gram-Schmidt Orthogonalization Consider the $2\times 2$ real matrix \[A=\begin{bmatrix} 1 & 1\\ 1& 3 \end{bmatrix}.\] (a) Prove that the matrix $A$ is positive definite. (b) Since $A$ is positive definite by part (a), the formula \[\langle \mathbf{x}, […]
- Determine a 2-Digit Number Satisfying Two Conditions A 2-digit number has two properties: The digits sum to 11, and if the number is written with digits reversed, and subtracted from the original number, the result is 45. Find the number. Solution. The key to this problem is noticing that our 2-digit number can be […]
- Finite 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 […]
- Find a Condition that a Vector be a Linear Combination Let \[\mathbf{v}=\begin{bmatrix} a \\ b \\ c \end{bmatrix}, \qquad \mathbf{v}_1=\begin{bmatrix} 1 \\ 2 \\ 0 \end{bmatrix}, \qquad \mathbf{v}_2=\begin{bmatrix} 2 \\ -1 \\ 2 \end{bmatrix}.\] Find the necessary and […]
- A Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable. Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]