# find-eigenvalues-eigenvectors

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• Eigenvalues of Real Skew-Symmetric Matrix are Zero or Purely Imaginary and the Rank is Even Let $A$ be a real skew-symmetric matrix, that is, $A^{\trans}=-A$. Then prove the following statements. (a) Each eigenvalue of the real skew-symmetric matrix $A$ is either $0$ or a purely imaginary number. (b) The rank of $A$ is even.   Proof. (a) Each […]
• Two Matrices with the Same Characteristic Polynomial. Diagonalize if Possible. Let $A=\begin{bmatrix} 1 & 3 & 3 \\ -3 &-5 &-3 \\ 3 & 3 & 1 \end{bmatrix} \text{ and } B=\begin{bmatrix} 2 & 4 & 3 \\ -4 &-6 &-3 \\ 3 & 3 & 1 \end{bmatrix}.$ For this problem, you may use the fact that both matrices have the same characteristic […]
• Construction of a Symmetric Matrix whose Inverse Matrix is Itself Let $\mathbf{v}$ be a nonzero vector in $\R^n$. Then the dot product $\mathbf{v}\cdot \mathbf{v}=\mathbf{v}^{\trans}\mathbf{v}\neq 0$. Set $a:=\frac{2}{\mathbf{v}^{\trans}\mathbf{v}}$ and define the $n\times n$ matrix $A$ by $A=I-a\mathbf{v}\mathbf{v}^{\trans},$ where […]
• The Center of a p-Group is Not Trivial Let $G$ be a group of order $|G|=p^n$ for some $n \in \N$. (Such a group is called a $p$-group.) Show that the center $Z(G)$ of the group $G$ is not trivial. Hint. Use the class equation. Proof. If $G=Z(G)$, then the statement is true. So suppose that $G\neq […] • 7 Problems on Skew-Symmetric Matrices Let$A$and$B$be$n\times n$skew-symmetric matrices. Namely$A^{\trans}=-A$and$B^{\trans}=-B$. (a) Prove that$A+B$is skew-symmetric. (b) Prove that$cA$is skew-symmetric for any scalar$c$. (c) Let$P$be an$m\times n$matrix. Prove that$P^{\trans}AP$is […] • Eigenvalues of a Stochastic Matrix is Always Less than or Equal to 1 Let$A=(a_{ij})$be an$n \times n$matrix. We say that$A=(a_{ij})$is a right stochastic matrix if each entry$a_{ij}$is nonnegative and the sum of the entries of each row is$1$. That is, we have $a_{ij}\geq 0 \quad \text{ and } \quad a_{i1}+a_{i2}+\cdots+a_{in}=1$ for$1 […]
• Exponential Functions are Linearly Independent Let $c_1, c_2,\dots, c_n$ be mutually distinct real numbers. Show that exponential functions $e^{c_1x}, e^{c_2x}, \dots, e^{c_nx}$ are linearly independent over $\R$. Hint. Consider a linear combination $a_1 e^{c_1 x}+a_2 e^{c_2x}+\cdots + a_ne^{c_nx}=0.$ […]
• Similar Matrices Have the Same Eigenvalues Show that if $A$ and $B$ are similar matrices, then they have the same eigenvalues and their algebraic multiplicities are the same. Proof. We prove that $A$ and $B$ have the same characteristic polynomial. Then the result follows immediately since eigenvalues and algebraic […]