Question

Explain why the $Q R$ algorithm fails to find the eigenvalues of the matrices (a) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$, (b) $\left(\begin{array}{rrr}-2 & 1 & 0 \\ 0 & -2 & 1 \\ 1 & 0 & -2\end{array}\right)$, (c) $\left(\begin{array}{rrr}5 & -4 & 2 \\ -4 & 5 & 2 \\ 2 & 2 & -1\end{array}\right)$.

    Explain why the $Q R$ algorithm fails to find the eigenvalues of the matrices
(a) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$,
(b) $\left(\begin{array}{rrr}-2 & 1 & 0 \\ 0 & -2 & 1 \\ 1 & 0 & -2\end{array}\right)$,
(c) $\left(\begin{array}{rrr}5 & -4 & 2 \\ -4 & 5 & 2 \\ 2 & 2 & -1\end{array}\right)$.

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Applied Linear Algebra (Undergraduate Texts in Mathematics)

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 9, Problem 15

Instant Answer

Step 1

It involves decomposing a matrix \( A \) into a product of an orthogonal matrix \( Q \) and an upper triangular matrix \( R \) (QR decomposition), and then forming a new matrix \( A_1 = RQ \). This process is repeated with \( A_1 \) to form \( A_2 \), and so on, Show more…

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Explain why the $Q R$ algorithm fails to find the eigenvalues of the matrices (a) $\left(\begin{array}{ll}0 & 1 \\ 1 & 0\end{array}\right)$, (b) $\left(\begin{array}{rrr}-2 & 1 & 0 \\ 0 & -2 & 1 \\ 1 & 0 & -2\end{array}\right)$, (c) $\left(\begin{array}{rrr}5 & -4 & 2 \\ -4 & 5 & 2 \\ 2 & 2 & -1\end{array}\right)$.

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