Question

Consider a vector p in 2-dimensional space. Let its direction (counter-clockwise angle with the positive x-axis) be θ. Let p be an eigenvector of a 2 × 2 matrix A with corresponding eigenvalue λ, λ > 0. If we denote the magnitude of a vector v by ||v||, identify the VALID statement regarding p', where p' = Ap.

A. Direction of p' = λθ, ||p'|| = λ ||p||
B. Direction of p' = θ, ||p'|| = ||p||/λ
C. Direction of p' = λθ, ||p'|| = ||p||
D. Direction of p' = θ, ||p'|| = λ||p||

Answer 1

The vector p', resulted from the multiplication of the eigenvector p by the matrix A, maintains the direction of p; its magnitude is scaled by the eigenvalue λ.

Step-by-step explanation:

When considering a vector p in 2-dimensional space that is an eigenvector of a 2 × 2 matrix A with corresponding positive eigenvalue λ, we observe specific transformations to p.

Since p is an eigenvector, after matrix multiplication by A, the resulting vector p' will maintain the same direction, θ, as the original vector p.

Moreover, its magnitude will be scaled by the corresponding eigenvalue λ, resulting in ||p'|| = λ ||p||.

The correct statement regarding the vector p', where p' = Ap, is therefore: Direction of p' = θ, and ||p'|| = λ||p||.

This means that the eigenvalue λ scales the magnitude of the eigenvector but does not affect its direction in 2-dimensional space.