mathematical equations

• 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 […]
• Compute the Product $A^{2017}\mathbf{u}$ of a Matrix Power and a Vector Let $A=\begin{bmatrix} -1 & 2 \\ 0 & -1 \end{bmatrix} \text{ and } \mathbf{u}=\begin{bmatrix} 1\\ 0 \end{bmatrix}.$ Compute $A^{2017}\mathbf{u}$.   (The Ohio State University, Linear Algebra Exam) Solution. We first compute $A\mathbf{u}$. We […]
• Companion Matrix for a Polynomial Consider a polynomial $p(x)=x^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0,$ where $a_i$ are real numbers. Define the matrix $A=\begin{bmatrix} 0 & 0 & \dots & 0 &-a_0 \\ 1 & 0 & \dots & 0 & -a_1 \\ 0 & 1 & \dots & 0 & -a_2 \\ \vdots & […] • 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 […] • Powers of a Diagonal Matrix Let A=\begin{bmatrix} a & 0\\ 0& b \end{bmatrix}. Show that (1) A^n=\begin{bmatrix} a^n & 0\\ 0& b^n \end{bmatrix} for any n \in \N. (2) Let B=S^{-1}AS, where S be an invertible 2 \times 2 matrix. Show that B^n=S^{-1}A^n S for any n \in […] • Find All Eigenvalues and Corresponding Eigenvectors for the 3\times 3 matrix Find all eigenvalues and corresponding eigenvectors for the matrix A if \[ A= \begin{bmatrix} 2 & -3 & 0 \\ 2 & -5 & 0 \\ 0 & 0 & 3 \end{bmatrix} .$   Solution. If $\lambda$ is an eigenvalue of $A$, then $\lambda$ […]