Deflation of the largest eigenvalue

Task number: 4372

Apply the theorem on the deflation of the largest eigenvalue to the matrix:

\(\begin{pmatrix} 1&2&1&2\\ 2&1&2&1\\ 1&2&1&2\\ 2&1&2&1 \end{pmatrix}\)
  • Řešení

    The matrix \(\mathbf A\) has eigenvalues 6, 0 (multiplicity two) nd \(-2\). The eigenvalue \(\lambda_1=6\) corresponds to eigenvector \(\mathbf x_1=(1{,}1,1{,}1)^T\), we normalize it to \(\mathbf z_1=\frac12(1{,}1,1{,}1)^T\).

    By the deflation we get the matrix \(\mathbf A'= \mathbf A - \lambda_1 \mathbf z_1 \mathbf z_1^T = \begin{pmatrix} 1&2&1&2\\ 2&1&2&1\\ 1&2&1&2\\ 2&1&2&1 \end{pmatrix} - 6\cdot \frac12 \begin{pmatrix} 1\\ 1\\ 1\\ 1 \end{pmatrix} \cdot \frac12 \begin{pmatrix} 1&1&1&1 \end{pmatrix} =\frac12 \begin{pmatrix} -1&1&-1&1\\ 1&-1&1&-1\\ -1&1&-1&1\\ 1&-1&1&-1 \end{pmatrix}\)

    The matrix \(\mathbf A'\) has eigenvalues 0 of multiplicity 3 and \(-2\) and the same eigenvectors as the matrix \(\mathbf A\).

Difficulty level: Moderate task
Routine calculation training
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