Numerical Methods · Unit 8
Eigenvalues and Eigenvectors
Exam-focused notes for Eigenvalues and Eigenvectors (Numerical Methods, BIT203): what the TU syllabus asks and how it has actually been tested, with 4 solved past questions from this unit.
What this unit covers
- Eigenvalue and eigenvector definitions
- Characteristic equation and determinant method
- Computing eigenvalues and eigenvectors
- Applications of eigenvalue problems
Eigenvalue and eigenvector definitions
Define eigenvalue and eigenvector. Distinguish between regression and interpolation. [2.5+2.5]
--- (a) Eigenvalue and Eigenvector Let $A$ be an $n \times n$ square matrix. A scalar $\lambda$ is called an eigenvalue of $A$ if there exists a non-zero vector $\mathbf{x}$ such that: $$A\mathbf{x} = \lambda\mathbf{x}$$ The non-zero vector $\mathbf{x}$ sat...
Full solved answer →Define eigen value and eigen vector. Explain how shooting method is used to solve boundary value problem. [5]
--- Definition: Let $A$ be an $n \times n$ square matrix. A scalar $\lambda$ is called an eigenvalue of $A$ if there exists a non-zero vector $\mathbf{x}$ such that: $$A\mathbf{x} = \lambda \mathbf{x}$$ The non-zero vector $\mathbf{x}$ satisfying this equat...
Full solved answer →Computing eigenvalues and eigenvectors
Find the Eigen values and Eigen vectors of the Matrix: $A=\begin{bmatrix} 3 & -1 \ 1 & 1 \end{bmatrix}$ [5]
Matrix: $$A = \begin{bmatrix} 3 & -1 \\ 1 & 1 \end{bmatrix}$$ Required: eigenvalues and eigenvectors. $$\det(A - \lambda I) = 0$$ $$A - \lambda I = \begin{bmatrix} 3-\lambda & -1 \\ 1 & 1-\lambda \end{bmatrix}$$ $$\det(A - \lambda I) = (3-\lambda)(1-\lambda...
Full solved answer →Determine the Eigen Values and corresponding Eigen Vectors for the matrix.
$$A = \begin{bmatrix} 1 & 6 & 1 \ 1 & 2 & 0 \ 0 & 0 & 3 \end{bmatrix}$$
[5]
$$A = \begin{bmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{bmatrix}$$ --- $$\det(A - \lambda I) = 0$$ $$A - \lambda I = \begin{bmatrix} 1-\lambda & 6 & 1 \\ 1 & 2-\lambda & 0 \\ 0 & 0 & 3-\lambda \end{bmatrix}$$ Expand along Row 3 (only one nonzero entr...
Full solved answer →Make Unit 8 stick
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