# univeristy-of-tokyo-eye-catch

by Yu · Published · Updated

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- Nilpotent Matrix and Eigenvalues of the Matrix An $n\times n$ matrix $A$ is called nilpotent if $A^k=O$, where $O$ is the $n\times n$ zero matrix. Prove the followings. (a) The matrix $A$ is nilpotent if and only if all the eigenvalues of $A$ is zero. (b) The matrix $A$ is nilpotent if and only if […]
- The Formula for the Inverse Matrix of $I+A$ for a $2\times 2$ Singular Matrix $A$ Let $A$ be a singular $2\times 2$ matrix such that $\tr(A)\neq -1$ and let $I$ be the $2\times 2$ identity matrix. Then prove that the inverse matrix of the matrix $I+A$ is given by the following formula: \[(I+A)^{-1}=I-\frac{1}{1+\tr(A)}A.\] Using the formula, calculate […]
- The Length of a Vector is Zero if and only if the Vector is the Zero Vector Let $\mathbf{v}$ be an $n \times 1$ column vector. Prove that $\mathbf{v}^\trans \mathbf{v} = 0$ if and only if $\mathbf{v}$ is the zero vector $\mathbf{0}$. Proof. Let $\mathbf{v} = \begin{bmatrix} v_1 \\ v_2 \\ \vdots \\ v_n \end{bmatrix} $. Then we […]
- Basis For Subspace Consisting of Matrices Commute With a Given Diagonal Matrix Let $V$ be the vector space of all $3\times 3$ real matrices. Let $A$ be the matrix given below and we define \[W=\{M\in V \mid AM=MA\}.\] That is, $W$ consists of matrices that commute with $A$. Then $W$ is a subspace of $V$. Determine which matrices are in the subspace $W$ […]
- Subspace of Skew-Symmetric Matrices and Its Dimension Let $V$ be the vector space of all $2\times 2$ matrices. Let $W$ be a subset of $V$ consisting of all $2\times 2$ skew-symmetric matrices. (Recall that a matrix $A$ is skew-symmetric if $A^{\trans}=-A$.) (a) Prove that the subset $W$ is a subspace of $V$. (b) Find the […]
- Rank of the Product of Matrices $AB$ is Less than or Equal to the Rank of $A$ Let $A$ be an $m \times n$ matrix and $B$ be an $n \times l$ matrix. Then prove the followings. (a) $\rk(AB) \leq \rk(A)$. (b) If the matrix $B$ is nonsingular, then $\rk(AB)=\rk(A)$. Hint. The rank of an $m \times n$ matrix $M$ is the dimension of the range […]
- Examples of Prime Ideals in Commutative Rings that are Not Maximal Ideals Give an example of a commutative ring $R$ and a prime ideal $I$ of $R$ that is not a maximal ideal of $R$. Solution. We give several examples. The key facts are: An ideal $I$ of $R$ is prime if and only if $R/I$ is an integral domain. An ideal $I$ of […]
- Symmetric Matrix and Its Eigenvalues, Eigenspaces, and Eigenspaces Let $A$ be a $4\times 4$ real symmetric matrix. Suppose that $\mathbf{v}_1=\begin{bmatrix} -1 \\ 2 \\ 0 \\ -1 \end{bmatrix}$ is an eigenvector corresponding to the eigenvalue $1$ of $A$. Suppose that the eigenspace for the eigenvalue $2$ is $3$-dimensional. (a) Find an […]