(a) Find the eigenvalues of the rotation matrix Rθ=( cosθ -sinθ sinθ cosθ ). For wh

step 1 Hello, welcome to problem 10 in eigenvalues and eigenvectors. We have this nice vector here, cos

(a) Find the eigenvalues of the rotation matrix Rθ=( ...

(a) Find the eigenvalues of the rotation matrix $R_\theta=\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$. For what values of $\theta$ are the eigenvalues real? (b) Explain why your answer gives an immediate solution to Exercise 1.5.7c.

(a) Find the eigenvalues of the rotation matrix $R_\theta=\left(\begin{array}{rr}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta\end{array}\right)$. For what values of $\theta$ are the eigenvalues real?
(b) Explain why your answer gives an immediate solution to Exercise 1.5.7c.

Key Concepts

Rotation Matrices

A rotation matrix is a linear transformation that rotates vectors in a Euclidean space without changing their length. In two dimensions, the matrix is structured with sine and cosine functions, and its properties include preserving angles and distances, which underlies many applications in geometry and physics.

Eigenvalues and the Characteristic Polynomial

Finding eigenvalues of a matrix involves constructing its characteristic polynomial by subtracting a scalar ? times the identity matrix from the given matrix and setting the determinant to zero. This polynomial reveals the scalars (eigenvalues) for which there exist nontrivial solutions to the corresponding homogeneous system. This concept is fundamental in understanding the behavior of linear transformations.

Real Versus Complex Eigenvalues

The condition under which a rotation matrix yields real eigenvalues is an essential concept in linear algebra. Generally, the eigenvalues of a two-dimensional rotation matrix are complex, unless the rotation angle is such that the associated trigonometric functions produce a discriminant leading to purely real solutions. This directly showcases the interplay between the algebraic nature of the characteristic polynomial and the geometric intuition behind rotations.

Application to Related Exercises

Connecting the eigenvalue analysis of a rotation matrix to other exercises, such as the one referenced, highlights how understanding the structure of specific matrices immediately provides insight into broader problem sets. By recognizing patterns and leveraging the properties of rotations and eigenvalues, one can often directly infer solutions to related problems without redundant computations.

Transcript

00:01 Hello, welcome to problem 10 in eigenvalues and eigenvectors.

00:04 We have this nice vector here, cosine theta, minus sine theta, sine theta, cosine theta.

00:17 And the target is to find the eigenvalues and eigenvectors.

00:21 We start usually by the eigenvalues by getting the determinant of this matrix, cosine theta, minus lambda, minus sine theta, sine theta and here we have cosine theta minus lambda as well okay and we make it equal to zero so in that case we have cosine theta minus lambda multiplied by cosine theta minus lambda and all of this is added to sine theta squared so from here we have cosine theta squared so from here we have cosine theta squared plus lambda squared minus 2, cosine theta lambda plus sine theta squared is equal to zero.

01:17 Cosign theta squared plus sine theta squared, this famous trinometric identity is equal to 1.

01:25 So we have lambda squared minus 2, cosine theta, lambda plus 1, is equal to 0.

01:33 The solution to this, there is like many algebraic ways to do it, but to keep it short, lambda 1 is simply cosine theta minus i sine theta, and lambda 2 is equal to cosine theta plus i sine theta.

01:53 So these are the eigenvalues and the target is to find the eigenvector corresponding to each one of them.

02:00 So let's start by lambda 1.

02:03 Lambda 1.

02:05 In the case of lambda 1, we find that the equation that we usually solve in that case, cosine theta minus lambda.

02:14 So that's the lambda here.

02:16 We see that the cosines cancelled and we are kept with i sine theta here.

02:24 And the rest of the matrix is the same.

02:27 The off diagonals does not change.

02:29 And this is also i.

02:32 Sine theta.

02:34 And if we multiply this by x1, x2, we find that we have the zero vector...