IGNOU BMTE 144 2026 - Determine the largest eigenvalue in magnitude and the correspond

Question

Determine the largest eigenvalue in magnitude and the corresponding eigenvector of the matrix $\begin{pmatrix} 1 & 6 & 1 \\ 1 & 2 & 0 \\ 0 & 0 & 3 \end{pmatrix}$ using the power method. Take (1, 0, 0)^T as the initial approximation and perform 4 iterations.

09 Jan 2026

Answer :

Word Count : 457

Let ( A = \begin{pmatrix} 1 & 6 & 1 \ 1 & 2 & 0 \ 0 & 0 & 3 \end{pmatrix} ) and the initial vector ( x^{(0)} = \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} ). We will perform 4 iterations of the power method to approximate the largest eigenvalue and its eigenvector. Iteration 1: [ x^{(1)} = A x^{(0)} = \begin{pmatrix} 1 & 6 & 1 \ 1 & 2 & 0 \ 0 & 0 & 3 \end{pmatrix} \begin{pmatrix} 1 \ 0 \ 0 \end{pmatrix} = \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix} ] Normalize ( x^{(1)} ) by its largest component (here, 1): [ x^{(1)} \approx \begin{pmatrix} 1 \ 1 \ 0 \end{pmatrix} ] Iteration 2: [ x^{(2)} = A x^{(1)} = \begin{pmatrix} 1 & 6 &

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