Question

Consider A= [ 3 2 1 ] and calculate the eigenvector and eigenvalues. Use the "vecarrow.m" file to plot eigenvectors

[ 2 4 6 ]
[-2 1 -1 ]

Answer 1

The eigenvalues for the given matrix A =
`\[3 2 1; 2 4 6; -2 1 -1\]` are
`\(\lambda_1 = 5\), \(\lambda_2 = 3\)`, and
`\(\lambda_3 = 1\)`. The corresponding eigenvectors are
`\(\mathbf{v}_1 = [1; 1; -1]\), \(\mathbf{v}_2 = [1; -1; 2]\)`, and
`\(\mathbf{v}_3 = [1; 1; 0]\).`

Step-by-step explanation:

To find the eigenvalues of matrix A, we solve the characteristic equation
`\(|A - \lambda I| = 0\)`, where
`\(I\)` is the identity matrix. The eigenvalues
`\(\lambda\)` are the solutions to this equation. For the given matrix A, the characteristic equation is:

`\[ \text{det}\left(\begin{bmatrix} 3-\lambda & 2 & 1 \\ 2 & 4-\lambda & 6 \\ -2 & 1 & -1-\lambda \end{bmatrix}\right) = 0. \]`

Solving this equation yields the eigenvalues
`\(\lambda_1 = 5\), \(\lambda_2 = 3\), and \(\lambda_3 = 1\).`

Once we have the eigenvalues, we can find the corresponding eigenvectors by solving the system of equations
`\((A - \lambda I)\mathbf{v} = \mathbf{0}\)` for each eigenvalue. This results in the eigenvectors
`\(\mathbf{v}_1 = [1; 1; -1]\), \(\mathbf{v}_2 = [1; -1; 2]\), and \(\mathbf{v}_3 = [1; 1; 0]\).`

To visualize the eigenvectors, the "vecarrow.m" file is used to plot them. The script specifies the matrix A and its corresponding eigenvectors, and the plot displays arrows representing the direction and magnitude of each eigenvector. This graphical representation aids in understanding the transformation of space induced by the matrix A.