A :
The top matrix on the menu is the diagonal matrix Initially, when you select this matrix, the vectors
x and Ax should both be aligned along the positive
x-axis. What information about an eigenvalue–
eigenvector pair is apparent from the initial figure
positions? Explain. Rotate x counterclockwise until
x and Ax are parallel, that is, until they both
lie along the same line through the origin. What
can you conclude about the second eigenvalue–
eigenvector pair? Repeat this experiment with the
second matrix. How can you determine the eigenvalues
and eigenvectors of a 2 × 2 diagonal matrix
by inspection without doing any computations?
Does this also work for 3 × 3 diagonal matrices?
Explain.
First, let's define the matrix A in question:
A = [[5/4, 0], [0, 3/4]]
We are told that the top matrix on the menu is the diagonal matrix, and that when we select this matrix, both the vector x and its image under A, Ax, are aligned along the positive x-axis. This means that x is already an eigenvector of A, with eigenvalue given by the diagonal entry corresponding to x. In this case, the matrix is already diagonal, so it is clear that the eigenvalues are 5/4 and 3/4, and the eigenvectors are the standard basis vectors [1, 0] and [0, 1].
Next, the question asks us to rotate the vector x counterclockwise until it and its image under A are parallel. To do this, we need to apply a rotation matrix to x, which will change its direction but not its magnitude.
The rotation matrix that rotates a vector by an angle θ counterclockwise can be defined as:
R(θ) = [[cos(θ), -sin(θ)], [sin(θ), cos(θ)]]
In this case, we want to rotate x by an angle of -45 degrees, so we can use the rotation matrix R(-45). We apply this matrix to the vector x to get the new vector x':
x' = R(-45) * x
= [[cos(-45), -sin(-45)], [sin(-45), cos(-45)]] * [1, 0]
= [sqrt(2)/2, -sqrt(2)/2]
We can now calculate Ax' to find the corresponding eigenvector of A:
Ax' = A * x'
= [[5/4, 0], [0, 3/4]] * [sqrt(2)/2, -sqrt(2)/2]
= [sqrt(2)/2, -sqrt(2)/2] * 5/4
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