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Nilpotent Matrix Problems and Solutions


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  • Sequence Converges to the Largest Eigenvalue of a MatrixSequence Converges to the Largest Eigenvalue of a Matrix Let $A$ be an $n\times n$ matrix. Suppose that $A$ has real eigenvalues $\lambda_1, \lambda_2, \dots, \lambda_n$ with corresponding eigenvectors $\mathbf{u}_1, \mathbf{u}_2, \dots, \mathbf{u}_n$. Furthermore, suppose that \[|\lambda_1| > |\lambda_2| \geq \cdots \geq […]
  • Example of an Infinite Algebraic ExtensionExample of an Infinite Algebraic Extension Find an example of an infinite algebraic extension over the field of rational numbers $\Q$ other than the algebraic closure $\bar{\Q}$ of $\Q$ in $\C$.   Definition (Algebraic Element, Algebraic Extension). Let $F$ be a field and let $E$ be an extension of […]
  • Quiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null SpaceQuiz 6. Determine Vectors in Null Space, Range / Find a Basis of Null Space (a) Let $A=\begin{bmatrix} 1 & 2 & 1 \\ 3 &6 &4 \end{bmatrix}$ and let \[\mathbf{a}=\begin{bmatrix} -3 \\ 1 \\ 1 \end{bmatrix}, \qquad \mathbf{b}=\begin{bmatrix} -2 \\ 1 \\ 0 \end{bmatrix}, \qquad \mathbf{c}=\begin{bmatrix} 1 \\ 1 […]
  • Boolean Rings Do Not Have Nonzero Nilpotent ElementsBoolean Rings Do Not Have Nonzero Nilpotent Elements Let $R$ be a commutative ring with $1$ such that every element $x$ in $R$ is idempotent, that is, $x^2=x$. (Such a ring is called a Boolean ring.) (a) Prove that $x^n=x$ for any positive integer $n$. (b) Prove that $R$ does not have a nonzero nilpotent […]
  • A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$A Linear Transformation $T: U\to V$ cannot be Injective if $\dim(U) > \dim(V)$ Let $U$ and $V$ be finite dimensional vector spaces over a scalar field $\F$. Consider a linear transformation $T:U\to V$. Prove that if $\dim(U) > \dim(V)$, then $T$ cannot be injective (one-to-one).   Hints. You may use the folowing facts. A linear […]
  • Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$Diagonalize the Complex Symmetric 3 by 3 Matrix with $\sin x$ and $\cos x$ Consider the complex matrix \[A=\begin{bmatrix} \sqrt{2}\cos x & i \sin x & 0 \\ i \sin x &0 &-i \sin x \\ 0 & -i \sin x & -\sqrt{2} \cos x \end{bmatrix},\] where $x$ is a real number between $0$ and $2\pi$. Determine for which values of $x$ the […]
  • Transpose of a Matrix and Eigenvalues and Related QuestionsTranspose of a Matrix and Eigenvalues and Related Questions Let $A$ be an $n \times n$ real matrix. Prove the followings. (a) The matrix $AA^{\trans}$ is a symmetric matrix. (b) The set of eigenvalues of $A$ and the set of eigenvalues of $A^{\trans}$ are equal. (c) The matrix $AA^{\trans}$ is non-negative definite. (An $n\times n$ […]
  • $\sqrt[m]{2}$ is an Irrational Number$\sqrt[m]{2}$ is an Irrational Number Prove that $\sqrt[m]{2}$ is an irrational number for any integer $m \geq 2$.   Hint. Use ring theory: Consider the polynomial $f(x)=x^m-2$. Apply Eisenstein's criterion, show that $f(x)$ is irreducible over $\Q$. Proof. Consider the monic polynomial […]

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