# Nagoya-university-eye-catch

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• There is at Least One Real Eigenvalue of an Odd Real Matrix Let $n$ be an odd integer and let $A$ be an $n\times n$ real matrix. Prove that the matrix $A$ has at least one real eigenvalue.   We give two proofs. Proof 1. Let $p(t)=\det(A-tI)$ be the characteristic polynomial of the matrix $A$. It is a degree $n$ […]
• A Recursive Relationship for a Power of a Matrix Suppose that the $2 \times 2$ matrix $A$ has eigenvalues $4$ and $-2$. For each integer $n \geq 1$, there are real numbers $b_n , c_n$ which satisfy the relation $A^{n} = b_n A + c_n I ,$ where $I$ is the identity matrix. Find $b_n$ and $c_n$ for $2 \leq n \leq 5$, and […]
• Find a Formula for a Linear Transformation If $L:\R^2 \to \R^3$ is a linear transformation such that \begin{align*} L\left( \begin{bmatrix} 1 \\ 0 \end{bmatrix}\right) =\begin{bmatrix} 1 \\ 1 \\ 2 \end{bmatrix}, \,\,\,\, L\left( \begin{bmatrix} 1 \\ 1 \end{bmatrix}\right) =\begin{bmatrix} 2 \\ 3 […]
• The Subspace of Linear Combinations whose Sums of Coefficients are zero Let $V$ be a vector space over a scalar field $K$. Let $\mathbf{v}_1, \mathbf{v}_2, \dots, \mathbf{v}_k$ be vectors in $V$ and consider the subset $W=\{a_1\mathbf{v}_1+a_2\mathbf{v}_2+\cdots+ a_k\mathbf{v}_k \mid a_1, a_2, \dots, a_k \in K \text{ and } […] • Is an Eigenvector of a Matrix an Eigenvector of its Inverse? Suppose that A is an n \times n matrix with eigenvalue \lambda and corresponding eigenvector \mathbf{v}. (a) If A is invertible, is \mathbf{v} an eigenvector of A^{-1}? If so, what is the corresponding eigenvalue? If not, explain why not. (b) Is 3\mathbf{v} an […] • A Homomorphism from the Additive Group of Integers to Itself Let \Z be the additive group of integers. Let f: \Z \to \Z be a group homomorphism. Then show that there exists an integer a such that \[f(n)=an$ for any integer $n$.   Hint. Let us first recall the definition of a group homomorphism. A group homomorphism from a […]
• The Center of the Heisenberg Group Over a Field $F$ is Isomorphic to the Additive Group $F$ Let $F$ be a field and let $H(F)=\left\{\, \begin{bmatrix} 1 & a & b \\ 0 &1 &c \\ 0 & 0 & 1 \end{bmatrix} \quad \middle| \quad \text{ for any} a,b,c\in F\, \right\}$ be the Heisenberg group over $F$. (The group operation of the Heisenberg group is matrix […]
• Given the Data of Eigenvalues, Determine if the Matrix is Invertible In each of the following cases, can we conclude that $A$ is invertible? If so, find an expression for $A^{-1}$ as a linear combination of positive powers of $A$. If $A$ is not invertible, explain why not. (a) The matrix $A$ is a $3 \times 3$ matrix with eigenvalues \$\lambda=i , […]