Question

TRUE OR FALSE

If $\mathbf{A}\mathbf{=}\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$then the matrix Q in the QR factorization of A is a rotation matrix.

Short answer

The given statement is false.

Step-by-step solution

Given information

Consider the given statement as follows.

If $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$, then matrix Q in the QR factorization of A is a rotation matrix.

Check the given statement

Consider the given matrix.

$A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$

To find the QR factorization of role="math" localid="1659626163025" $A=\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$

A direct computation or geometric approach shows that$Q=\frac{1}{\sqrt{5}}\left[\begin{array}{cc}1& 2\\ 2& 1\end{array}\right]$represents a matrix, not a rotation matrix.

This can be obtained geometrically as the vector perpendicular to the vector$\left[\begin{array}{c}1\\ 2\end{array}\right]$is either $\left[\begin{array}{c}2\\ 1\end{array}\right]$or $\left[\begin{array}{c}-2\\ 1\end{array}\right]$.

Hence, the given statement “The matrix Q in the QR factorization of A is a rotation matrix is false”.