#fourier

Question 1: Single choice

hardAI-checkedConvolutional Neural Networks

Every $n \times n$ circulant matrix is a polynomial $C = c_0 I + c_1 P + \dots + c_{n-1}P^{n-1}$ in the cyclic shift $P$ , so it performs a cyclic convolution. What are the eigenvectors of circulant matrices, and what follows for the product $CD$ of two circulants?

The Fourier modes; the eigenvalues of $CD$ are the products of those of $C$ and $D$ 1
The standard basis vectors; the eigenvalues of $CD$ are the products of those of $C$ and $D$ 2
The Fourier modes; the eigenvalues of $CD$ are the sums of those of $C$ and $D$ 3
They depend on the coefficients $c_k$ , so two circulants share eigenvectors only if $CD = DC$ 4

Timed test