Mathematics II · Unit 7 · 5 hrs
Eigenvalues and Eigen Vectors
Exam-focused notes for Eigenvalues and Eigen Vectors (Mathematics II, MTH168): what the TU syllabus asks and how it has actually been tested, with 7 solved past questions from this unit.
What this unit covers
- Eigenvectors and Eigenvalues
- The characteristic equations
- Diagonalization
- Eigenvectors and linear transformations
- Complex eigenvalues
- Discrete dynamical systems
- Applications to differential equations
The characteristic equations
Find the eigenvalue of $A = \begin{bmatrix} 7 & 3 \ 3 & -1 \end{bmatrix}$ [5]
$$A = \begin{bmatrix} 7 & 3 \\ 3 & -1 \end{bmatrix}$$ --- $$\det(A - \lambda I) = 0$$ $$A - \lambda I = \begin{bmatrix} 7 - \lambda & 3 \\ 3 & -1 - \lambda \end{bmatrix}$$ $$\det(A - \lambda I) = (7 - \lambda)(-1 - \lambda) - (3)(3)$$ Expand: $$(7 - \lambda...
Full solved answer →Find the eigenvalues of the matrix $$\begin{bmatrix} 6 & 3 & -8 \ 0 & -2 & 0 \ 1 & 0 & -3 \end{bmatrix}$$ [5]
$$A = \begin{bmatrix} 6 & 3 & -8 \\ 0 & -2 & 0 \\ 1 & 0 & -3 \end{bmatrix}$$ All entries are readable and complete. $$\det(A - \lambda I) = 0$$ $$A - \lambda I = \begin{bmatrix} 6-\lambda & 3 & -8 \\ 0 & -2-\lambda & 0 \\ 1 & 0 & -3-\lambda \end{bmatrix}$$ ...
Full solved answer →Find the eigenvalues of the matrix $$\begin{bmatrix} 6 & 3 & -8 \ 0 & -2 & 0 \ 1 & 0 & -3 \end{bmatrix}$$ [5]
$$A = \begin{bmatrix} 6 & 3 & -8 \\ 0 & -2 & 0 \\ 1 & 0 & -3 \end{bmatrix}$$ Eigenvalues satisfy $\det(A - \lambda I) = 0$. $$A - \lambda I = \begin{bmatrix} 6-\lambda & 3 & -8 \\ 0 & -2-\lambda & 0 \\ 1 & 0 & -3-\lambda \end{bmatrix}$$ Row 2 has entries $0...
Full solved answer →Eigenvectors and Eigenvalues
Find the eigen value of $$\begin{bmatrix} 3 & 6 & -8 \ 0 & 0 & 6 \ 0 & 0 & 2 \end{bmatrix}$$ [5]
Matrix: $$A = \begin{bmatrix} 3 & 6 & -8 \\ 0 & 0 & 6 \\ 0 & 0 & 2 \end{bmatrix}$$ Eigenvalues satisfy $\det(A - \lambda I) = 0$. $$A - \lambda I = \begin{bmatrix} 3-\lambda & 6 & -8 \\ 0 & -\lambda & 6 \\ 0 & 0 & 2-\lambda \end{bmatrix}$$ Since $A - \lambd...
Full solved answer →Find the eigen values of the matrix $\begin{bmatrix} 6 & 5 \ -8 & -6 \end{bmatrix}$ [5]
Given data: Matrix: $$A = \begin{bmatrix} 6 & 5 \\ -8 & -6 \end{bmatrix}$$ All numeric inputs present. No missing data. --- Characteristic equation: $\det(A - \lambda I) = 0$ $$A - \lambda I = \begin{bmatrix} 6 - \lambda & 5 \\ -8 & -6 - \lambda \end{bmatri...
Full solved answer →Is $\begin{bmatrix} 3 \ 2 \end{bmatrix}$ an eigen vector of $\begin{bmatrix} 5 & -3 \ -4 & 9 \end{bmatrix}$? If so, find eigenvalue. [5]
$$A = \begin{bmatrix} 5 & -3 \\ -4 & 9 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 3 \\ 2 \end{bmatrix}$$ Requirement: Determine whether $\mathbf{v}$ is an eigenvector of $A$; if so, find the eigenvalue. --- A non-zero vector $\mathbf{v}$ is an eigenv...
Full solved answer →Diagonalization
If $A=\begin{bmatrix} 7 & 2 \ -4 & 1 \end{bmatrix}$, find a formula for $A^n$, where $A = PDP^{-1}$, $P=\begin{bmatrix} 1 & 1 \ -1 & -2 \end{bmatrix}$ and $D=\begin{bmatrix} 5 & 0 \ 0 & 3 \end{bmatrix}$ [5]
Given data: $$A = \begin{bmatrix} 7 & 2 \\ -4 & 1 \end{bmatrix}, \quad P = \begin{bmatrix} 1 & 1 \\ -1 & -2 \end{bmatrix}, \quad D = \begin{bmatrix} 5 & 0 \\ 0 & 3 \end{bmatrix}$$ Relation: $A = PDP^{-1}$. Required: formula for $A^n$. Key fact: $A^n = (PDP^...
Full solved answer →Make Unit 7 stick
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