OpenStudy (datanewb):

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?

OpenStudy (anonymous):

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.

OpenStudy (anonymous):

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