A professor teaching linear algebra starts with a square matrix G whose QR decomposition is given by $G = Q_0R_0$

A professor teaching linear algebra starts with a square matrix G whose QR decomposition is given by $G = Q_0R_0$ . He then defines matrices $H = R_0Q_0$ which has a QR decomposition of the form $G = Q_1R_1$ and a matrix $M = R_1Q_1$ . Given this, he asks the students to answer the following:
i) Do the matrices G and H have the same eigenvalues? Justify.
ii) Prove of disprove that G and M have the same eigenvalues.

Solution

What we know from the question:

(1) $\begin{equation*} G = Q_0R_0 \end{equation*}$

(2) $\begin{equation*} H = R_0Q_0 \end{equation*}$

(3) $\begin{equation*} H = Q_1R_1 \end{equation*}$

(4) $\begin{equation*} M = R_1Q_1 \end{equation*}$

i) Consider (1), $G = Q_0R_0$
$Q_0$ is orthonormalized form of G.

(5) $\begin{equation*} \left\ R_0 = Q_0^TG \end{equation*}$

Apply (5) in (2) $=>$

(6) $\begin{equation*} \left\ H = Q_0^TGQ_0 \end{equation*}$

But since,

(7) $\begin{equation*} \left\ Q_0^T = Q_0^{-1} \end{equation*}$

Apply (7) in (6) $=>$

(8) $\begin{equation*} \left\ H = Q_0^{-1}GQ_0 \end{equation*}$

(8) is a form of diagonalization and implies that H and G are similar matrices. I.e. H and G have the same eigenvalues.

ii) Consider (3), , $H = Q_1R_1$
$Q_1$ is orthonormalized form of G. Thus,

(9) $\begin{equation*} \left\ R_1 = Q_1^TH \end{equation*}$

Apply (9) in (4) =>

(10) $\begin{equation*} \left\ M = Q_1^THQ_1 \end{equation*}$

(11) $\begin{equation*} \left\ M = Q_1^{-1}HQ_1 \end{equation*}$

(11) => M and H are similar and have the same eigenvalues. Combine this with (8) and it means that G and M have the same eigenvalues.

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