# decimal expansion of pi

### More from my site

• Commuting 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 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 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. 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 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 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 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 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 […]