Question

Apply the $Q R$ algorithm to the following non-symmetric matrices to find their eigenvalues to 3 decimal places: (a) $\left(\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right)$, (b) $\left(\begin{array}{ll}2 & 3 \\ 1 & 5\end{array}\right)$, (c) $\left(\begin{array}{rrr}2 & 1 & 0 \\ 2 & 0 & -3 \\ 0 & -2 & 1\end{array}\right)$, (d) $\left(\begin{array}{rrr}2 & 5 & 1 \\ 2 & -1 & 3 \\ 4 & 5 & 3\end{array}\right)$, (e) $\left(\begin{array}{rrrr}6 & 1 & 7 & 9 \\ 6 & 8 & 14 & 9 \\ 3 & 1 & 4 & 6 \\ 3 & 2 & 5 & 3\end{array}\right)$

    Apply the $Q R$ algorithm to the following non-symmetric matrices to find their eigenvalues to 3 decimal places:
(a) $\left(\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right)$,
(b) $\left(\begin{array}{ll}2 & 3 \\ 1 & 5\end{array}\right)$,
(c) $\left(\begin{array}{rrr}2 & 1 & 0 \\ 2 & 0 & -3 \\ 0 & -2 & 1\end{array}\right)$,
(d) $\left(\begin{array}{rrr}2 & 5 & 1 \\ 2 & -1 & 3 \\ 4 & 5 & 3\end{array}\right)$,
(e) $\left(\begin{array}{rrrr}6 & 1 & 7 & 9 \\ 6 & 8 & 14 & 9 \\ 3 & 1 & 4 & 6 \\ 3 & 2 & 5 & 3\end{array}\right)$

Applied Linear Algebra (Undergraduate Texts in Mathematics)

Peter J. Olver,… 2nd Edition

Chapter 9, Problem 13

Instant Answer

Step 1

It involves decomposing a matrix $ A $ into a product of an orthogonal matrix $ Q $ and an upper triangular matrix $ R $, and then forming a new matrix by multiplying $ R $ by $ Q $. This process is repeated with the new matrix, and as iterations Show more…

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Apply the $Q R$ algorithm to the following non-symmetric matrices to find their eigenvalues to 3 decimal places: (a) $\left(\begin{array}{rr}-1 & -2 \\ 3 & 4\end{array}\right)$, (b) $\left(\begin{array}{ll}2 & 3 \\ 1 & 5\end{array}\right)$, (c) $\left(\begin{array}{rrr}2 & 1 & 0 \\ 2 & 0 & -3 \\ 0 & -2 & 1\end{array}\right)$, (d) $\left(\begin{array}{rrr}2 & 5 & 1 \\ 2 & -1 & 3 \\ 4 & 5 & 3\end{array}\right)$, (e) $\left(\begin{array}{rrrr}6 & 1 & 7 & 9 \\ 6 & 8 & 14 & 9 \\ 3 & 1 & 4 & 6 \\ 3 & 2 & 5 & 3\end{array}\right)$

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