Is the maximum rotation by multiplying a positive definite matrix is less than 90 degrees?

1.4k Views Asked by At

To have intuition, I think of rotation of a vector $v$ in a two dimensional space. A vector can be rotated by multiplying the rotation matrix as $$ R=\begin{bmatrix} cos\theta & -sin\theta \\ sin\theta & cos\theta \\ \end{bmatrix} $$ However, the rotation matrix is positive definite only or $-\frac{\pi}{2}\leq \theta \leq \frac{\pi}{2}$.

So, if we impose the positive definiteness onto the rotation matrix, it just can rotate the vector at most 90 degrees (clockwise or counterclockwise).

Now, since every matrix multiplication has its rotational effect, can we conclude that the maximum rotation by multiplying a positive definite matrix is less than 90 degrees?

1

There are 1 best solutions below

0
On

Yes. Suppose A is a 2x2 positive definite matrix. Then for any non-zero x, $x' A x > 0$. But $x' A x = |x| |Ax| cos(\theta)$, for $\theta$ the angle between x and Ax, from which we see that $cos(\theta) > 0$ and thus $\theta$ must be between $-\pi/2$ and $\pi/2$.