# field-theory-2

by Yu · Published · Updated

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- Stochastic Matrix (Markov Matrix) and its Eigenvalues and Eigenvectors (a) Let \[A=\begin{bmatrix} a_{11} & a_{12}\\ a_{21}& a_{22} \end{bmatrix}\] be a matrix such that $a_{11}+a_{12}=1$ and $a_{21}+a_{22}=1$. Namely, the sum of the entries in each row is $1$. (Such a matrix is called (right) stochastic matrix (also termed […]
- Find All Matrices $B$ that Commutes With a Given Matrix $A$: $AB=BA$ Let \[A=\begin{bmatrix} 1 & 3\\ 2& 4 \end{bmatrix}.\] Then (a) Find all matrices \[B=\begin{bmatrix} x & y\\ z& w \end{bmatrix}\] such that $AB=BA$. (b) Use the results of part (a) to exhibit $2\times 2$ matrices $B$ and $C$ such that \[AB=BA \text{ and } […]
- Prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$ Using the Matrix Components Let $A$ and $B$ be $n \times n$ matrices, and $\mathbf{v}$ an $n \times 1$ column vector. Use the matrix components to prove that $(A + B) \mathbf{v} = A\mathbf{v} + B\mathbf{v}$. Solution. We will use the matrix components $A = (a_{i j})_{1 \leq i, j \leq n}$, $B = […]
- Rotation Matrix in the Plane and its Eigenvalues and Eigenvectors Consider the $2\times 2$ matrix \[A=\begin{bmatrix} \cos \theta & -\sin \theta\\ \sin \theta& \cos \theta \end{bmatrix},\] where $\theta$ is a real number $0\leq \theta < 2\pi$. (a) Find the characteristic polynomial of the matrix $A$. (b) Find the […]
- If the Images of Vectors are Linearly Independent, then They Are Linearly Independent Let $T: \R^n \to \R^m$ be a linear transformation. Suppose that $S=\{\mathbf{x}_1, \mathbf{x}_2,\dots, \mathbf{x}_k\}$ is a subset of $\R^n$ such that $\{T(\mathbf{x}_1), T(\mathbf{x}_2), \dots, T(\mathbf{x}_k) \}$ is a linearly independent subset of $\R^m$. Prove that the set $S$ […]
- If Two Ideals Are Comaximal in a Commutative Ring, then Their Powers Are Comaximal Ideals Let $R$ be a commutative ring and let $I_1$ and $I_2$ be comaximal ideals. That is, we have \[I_1+I_2=R.\] Then show that for any positive integers $m$ and $n$, the ideals $I_1^m$ and $I_2^n$ are comaximal. > Proof. Since $I_1+I_2=R$, there exists $a \in I_1$ […]
- If the Quotient Ring is a Field, then the Ideal is Maximal Let $R$ be a ring with unit $1\neq 0$. Prove that if $M$ is an ideal of $R$ such that $R/M$ is a field, then $M$ is a maximal ideal of $R$. (Do not assume that the ring $R$ is commutative.) Proof. Let $I$ be an ideal of $R$ such that \[M \subset I \subset […]
- Sherman-Woodbery Formula for the Inverse Matrix Let $\mathbf{u}$ and $\mathbf{v}$ be vectors in $\R^n$, and let $I$ be the $n \times n$ identity matrix. Suppose that the inner product of $\mathbf{u}$ and $\mathbf{v}$ satisfies \[\mathbf{v}^{\trans}\mathbf{u}\neq -1.\] Define the matrix […]