Question

In the lecture video on Eigenvalues (Lecture 21), at 31:00 Strang says the 2x2 rotation vector Q is \[Q = \begin{bmatrix}
\cos{90} & -\sin 90 \\
\sin 90& \cos 90
\end{bmatrix}\]. Why are cos and sin used as they are in this matrix?

Answer 1

In general, if you want to rotate a vector in R2 by theta, the rotation matrix is:\[\left(\begin{array} {rr} \cos\theta & -\sin\theta \\ \sin\theta & \cos\theta \end{array}\right)\]He just plugged in 90 because he wanted to represent a rotation by 90 degrees.

Answer 2

If you consider X as a vector \[\vec{i} + \vec{j}\] (where \[\vec{i}\ and\ \vec{j}\] are unit vectors along x and y directions respectively) then X can be written in matrice form as \[\left( \begin{array}{ccc} 1 \\ 1 \end{array} \right)\]
If you rotate this by 90 degrees then you will get the resulting vector as \[-\vec{i} + \vec{j}\]
which in matrice form would be \[\left( \begin{array}{ccc} 1 \\ -1 \end{array} \right)\]

Coming back to the question, Q is the operation matrice that rotates a vector X by 90 degrees. The resulting rotated vector would be QX which is \[\left( \begin{array}{ccc} 1 \\ -1 \end{array} \right)\] for the above case
Hope this helps