Householder Feedback Matrix

Jot proposed the lossless feedback matrix

$$\mathbf{A}_N = \mathbf{I}_N - \frac{2}{N}\underline{u}_N\underline{u}_N^T \protect$$

• Householder reflection (negated)
• Input vector is reflected about $\underline{u}_N^T = [1, 1, \dots, 1]$ in $N$ -dimensional space
• $\mathbf{I}_N$ can be replaced by any $N\times N$ permutation matrix
• Multiply-free when $N$ is a power of 2
• At most one multiply required
• Only $2N-1$ additions
  (a general matrix-times-vector multiplication is $O(N^2)$ multiply-adds)