decimal expansion of pi

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decimal expansion of pi


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  • Commuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same EigenvaluesCommuting Matrices $AB=BA$ such that $A-B$ is Nilpotent Have the Same Eigenvalues Let $A$ and $B$ be square matrices such that they commute each other: $AB=BA$. Assume that $A-B$ is a nilpotent matrix. Then prove that the eigenvalues of $A$ and $B$ are the same.   Proof. Let $N:=A-B$. By assumption, the matrix $N$ is nilpotent. This […]
  • A Matrix Having One Positive Eigenvalue and One Negative EigenvalueA Matrix Having One Positive Eigenvalue and One Negative Eigenvalue Prove that the matrix \[A=\begin{bmatrix} 1 & 1.00001 & 1 \\ 1.00001 &1 &1.00001 \\ 1 & 1.00001 & 1 \end{bmatrix}\] has one positive eigenvalue and one negative eigenvalue. (University of California, Berkeley Qualifying Exam Problem)   Solution. Let us put […]
  • Find All the Values of $x$ so that a Given $3\times 3$ Matrix is SingularFind All the Values of $x$ so that a Given $3\times 3$ Matrix is Singular Find all the values of $x$ so that the following matrix $A$ is a singular matrix. \[A=\begin{bmatrix} x & x^2 & 1 \\ 2 &3 &1 \\ 0 & -1 & 1 \end{bmatrix}.\]   Hint. Use the fact that a matrix is singular if and only if its determinant is […]
  • How to Find Eigenvalues of a Specific Matrix.How to Find Eigenvalues of a Specific Matrix. Find all eigenvalues of the following $n \times n$ matrix. \[ A=\begin{bmatrix} 0 & 0 & \cdots & 0 &1 \\ 1 & 0 & \cdots & 0 & 0\\ 0 & 1 & \cdots & 0 &0\\ \vdots & \vdots & \ddots & \ddots & \vdots \\ 0 & […]
  • The Rotation Matrix is an Orthogonal TransformationThe Rotation Matrix is an Orthogonal Transformation Let $\mathbb{R}^2$ be the vector space of size-2 column vectors. This vector space has an inner product defined by $ \langle \mathbf{v} , \mathbf{w} \rangle = \mathbf{v}^\trans \mathbf{w}$. A linear transformation $T : \R^2 \rightarrow \R^2$ is called an orthogonal transformation if […]
  • Mathematics About the Number 2018Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018.   Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is \[2018=2\cdot 1009.\] Here $2$ and $1009$ are […]
  • The Sum of Cosine Squared in an Inner Product SpaceThe Sum of Cosine Squared in an Inner Product Space Let $\mathbf{v}$ be a vector in an inner product space $V$ over $\R$. Suppose that $\{\mathbf{u}_1, \dots, \mathbf{u}_n\}$ is an orthonormal basis of $V$. Let $\theta_i$ be the angle between $\mathbf{v}$ and $\mathbf{u}_i$ for $i=1,\dots, n$. Prove that \[\cos […]
  • The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two ElementsThe Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements Let $G$ be an abelian group. Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively. Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$. Also determine whether the statement is true if $G$ is a […]

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