Chapter 4.1, Problem 38EQ

Find an eigenbasis and diagonalize. (Show the details.)

Pi- Determine which f th falining equations San an eigen funetion equation , and qive th valuet sinfIx) TTX a) 2. XP d sin(" dx2 iky

Convert the given quadratic form to canonical form by finding the eigen values and eigen vectors .

be the matrices C and D. Find the eigenvalues and eigenvectors of C and D.

Show all work and every step. That is the only way I will know how you solved the problem. Directions: Find the bases for the eigenspaces of A.

Find all of the eigenvalues of the matrix A over the complex numbers C. Give bases for each of the corresponding eigenspaces. A = 4-1-1 = 1+i has eigenspace span has eigenspace span × !- (2-value with smaller imaginary part) (2-value with larger imaginary part)

Determine the eigenvalues and eigenvectors of the following matrix: -3 6 -3 3 Enter X = a +3 i where 3 > 0. Enter the eigenvector as a vector with complex-valued entries. Note that if you found the eigenvector using ₂ = a-3. i then the desired vector is the complex conjugate of this vector. That is, you do not need to compute the eigenvector again! Separate the coordinates of the vectors with commas. λ = V

Diagonalize the matrix A = ㅕ -1 3-4 -2 3-4 1 1 3 Verify that AS = SA. Note that the some of the eigenval- ues and eigenvectors include complex numbers.

b. Draw the locus of the eigenvalues you found in part (a) in a complex plane. Indicate the location of roots for 0 = 30°, 45°, 90° . For which angles of 0 the eigenvalues of the rotation matrix are real (the imaginary part is zero)? Imaginary +i Real -1 C. Having the eigenvalues A1 and d2, find the complex eigenvectors of the rotation matrix. Note that the vector a where a and B are complex numbers and cos 0 – Ai - sin 0 +B Cos - is an sin 0 eigenvector for the eigenvalue of d;. Do eigenvectors of the rotation matrix depend on the angle of rotation 0?