Cayley-Hamilton

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Cayley-Hamilton Theorem Problems and Solutions


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  • Quiz 13 (Part 1) Diagonalize a MatrixQuiz 13 (Part 1) Diagonalize a Matrix Let \[A=\begin{bmatrix} 2 & -1 & -1 \\ -1 &2 &-1 \\ -1 & -1 & 2 \end{bmatrix}.\] Determine whether the matrix $A$ is diagonalizable. If it is diagonalizable, then diagonalize $A$. That is, find a nonsingular matrix $A$ and a diagonal matrix $D$ such that […]
  • Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam)Characteristic Polynomial, Eigenvalues, Diagonalization Problem (Princeton University Exam) Let \[\begin{bmatrix} 0 & 0 & 1 \\ 1 &0 &0 \\ 0 & 1 & 0 \end{bmatrix}.\] (a) Find the characteristic polynomial and all the eigenvalues (real and complex) of $A$. Is $A$ diagonalizable over the complex numbers? (b) Calculate $A^{2009}$. (Princeton University, […]
  • A Matrix Similar to a Diagonalizable Matrix is Also DiagonalizableA Matrix Similar to a Diagonalizable Matrix is Also Diagonalizable Let $A, B$ be matrices. Show that if $A$ is diagonalizable and if $B$ is similar to $A$, then $B$ is diagonalizable.   Definitions/Hint. Recall the relevant definitions. Two matrices $A$ and $B$ are similar if there exists a nonsingular (invertible) matrix $S$ such […]
  • Companion Matrix for a PolynomialCompanion 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 & […]
  • Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$Find a General Formula of a Linear Transformation From $\R^2$ to $\R^3$ Suppose that $T: \R^2 \to \R^3$ is a linear transformation satisfying \[T\left(\, \begin{bmatrix} 1 \\ 2 \end{bmatrix}\,\right)=\begin{bmatrix} 3 \\ 4 \\ 5 \end{bmatrix} \text{ and } T\left(\, \begin{bmatrix} 0 \\ 1 \end{bmatrix} […]
  • Quiz 7. Find a Basis of the Range, Rank, and Nullity of a MatrixQuiz 7. Find a Basis of the Range, Rank, and Nullity of a Matrix (a) Let $A=\begin{bmatrix} 1 & 3 & 0 & 0 \\ 1 &3 & 1 & 2 \\ 1 & 3 & 1 & 2 \end{bmatrix}$. Find a basis for the range $\calR(A)$ of $A$ that consists of columns of $A$. (b) Find the rank and nullity of the matrix $A$ in part (a).   Solution. (a) […]
  • Express a Vector as a Linear Combination of Given Three VectorsExpress a Vector as a Linear Combination of Given Three Vectors Let \[\mathbf{v}_1=\begin{bmatrix} 1 \\ 5 \\ -1 \end{bmatrix}, \mathbf{v}_2=\begin{bmatrix} 1 \\ 4 \\ 3 \end{bmatrix}, \mathbf{v}_3=\begin{bmatrix} 1 \\ 2 \\ 1 \end{bmatrix}, \mathbf{b}=\begin{bmatrix} 2 \\ 13 \\ 6 […]
  • Null Space, Nullity, Range, Rank of a Projection Linear TransformationNull Space, Nullity, Range, Rank of a Projection Linear Transformation Let $\mathbf{u}=\begin{bmatrix} 1 \\ 1 \\ 0 \end{bmatrix}$ and $T:\R^3 \to \R^3$ be the linear transformation \[T(\mathbf{x})=\proj_{\mathbf{u}}\mathbf{x}=\left(\, \frac{\mathbf{u}\cdot \mathbf{x}}{\mathbf{u}\cdot \mathbf{u}} \,\right)\mathbf{u}.\] (a) […]

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