# abelian-group-eye-catch

by Yu · Published · Updated

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- Orthogonality of Eigenvectors of a Symmetric Matrix Corresponding to Distinct Eigenvalues Suppose that a real symmetric matrix $A$ has two distinct eigenvalues $\alpha$ and $\beta$. Show that any eigenvector corresponding to $\alpha$ is orthogonal to any eigenvector corresponding to $\beta$. (Nagoya University, Linear Algebra Final Exam Problem) Hint. Two […]
- A Square Root Matrix of a Symmetric Matrix Answer the following two questions with justification. (a) Does there exist a $2 \times 2$ matrix $A$ with $A^3=O$ but $A^2 \neq O$? Here $O$ denotes the $2 \times 2$ zero matrix. (b) Does there exist a $3 \times 3$ real matrix $B$ such that $B^2=A$ […]
- A Ring is Commutative if Whenever $ab=ca$, then $b=c$ Let $R$ be a ring and assume that whenever $ab=ca$ for some elements $a, b, c\in R$, we have $b=c$. Then prove that $R$ is a commutative ring. Proof. Let $x, y$ be arbitrary elements in $R$. We want to show that $xy=yx$. Consider the […]
- Every Group of Order 12 Has a Normal Subgroup of Order 3 or 4 Let $G$ be a group of order $12$. Prove that $G$ has a normal subgroup of order $3$ or $4$. Hint. Use Sylow's theorem. (See Sylow’s Theorem (Summary) for a review of Sylow's theorem.) Recall that if there is a unique Sylow $p$-subgroup in a group $GH$, then it is […]
- Galois Extension $\Q(\sqrt{2+\sqrt{2}})$ of Degree 4 with Cyclic Group Show that $\Q(\sqrt{2+\sqrt{2}})$ is a cyclic quartic field, that is, it is a Galois extension of degree $4$ with cyclic Galois group. Proof. Put $\alpha=\sqrt{2+\sqrt{2}}$. Then we have $\alpha^2=2+\sqrt{2}$. Taking square of $\alpha^2-2=\sqrt{2}$, we obtain […]
- In which $\R^k$, are the Nullspace and Range Subspaces? Let $A$ be an $m \times n$ matrix. Suppose that the nullspace of $A$ is a plane in $\R^3$ and the range is spanned by a nonzero vector $\mathbf{v}$ in $\R^5$. Determine $m$ and $n$. Also, find the rank and nullity of $A$. Solution. For an $m \times n$ matrix $A$, the […]
- 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 […]
- If a Matrix $A$ is Singular, then Exists Nonzero $B$ such that $AB$ is the Zero Matrix Let $A$ be a $3\times 3$ singular matrix. Then show that there exists a nonzero $3\times 3$ matrix $B$ such that \[AB=O,\] where $O$ is the $3\times 3$ zero matrix. Proof. Since $A$ is singular, the equation $A\mathbf{x}=\mathbf{0}$ has a nonzero […]