# mathematical equations

• 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 […] • Diagonalize the 2\times 2 Hermitian Matrix by a Unitary Matrix Consider the Hermitian matrix \[A=\begin{bmatrix} 1 & i\\ -i& 1 \end{bmatrix}.$ (a) Find the eigenvalues of $A$. (b) For each eigenvalue of $A$, find the eigenvectors. (c) Diagonalize the Hermitian matrix $A$ by a unitary matrix. Namely, find a diagonal matrix […]
• Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are \$n\times […]