Householder Properties for Specific Sizes

• For $N\neq 2$ , all entries in the matrix are nonzero
  • Every delay line feeds back to every other delay line
  • Echo density maximized as soon as possible
• For $N=4$ , all matrix entries have the same magnitude:
  $$\mathbf{A}_4 = \frac{1}{2} \left[\begin{array}{rrrr} 1 & -1 & -1 ... ...\ -1 & 1 & -1 & -1\\ -1 & -1 & 1 & -1\\ -1 & -1 & -1 & 1 \end{array}\right]$$
  .
• Only the $N=4$ case is ``balanced'' in this way
• Multiply free
• In a manner analogous to Hadamard embedding to generate higher-order Hadamard matrices, Jot proposed constructing an $N=16$ feedback matrix as a $4\times 4$ Householder embedding of the $N=4$ Householder matrix:
  $$\mathbf{A}_{16} = \frac{1}{2} \left[\begin{array}{rrrr} \mathbf{A... ...\mathbf{A}_4 & -\mathbf{A}_4 & -\mathbf{A}_4 & \mathbf{A}_4 \end{array}\right]$$

Triangular Feedback Matrices

A triangular matrix has its eigenvalues along the diagonal.
Example:

$$\mathbf{A}_3 = \left[\begin{array}{ccc} \lambda_1 & 0 & 0\\ [2pt] a & \lambda_2 & 0\\ [2pt] b & c & \lambda_3 \end{array}\right]$$

is lower triangular. Its eigenvalues are $(\lambda_1, \lambda_2,\lambda_3)$ for all $a$ , $b$ , $c$ .

Note: Not all triangular matrices with unit-modulus eigenvalues are lossless.
Example:

$$\mathbf{A}_2 = \left[\begin{array}{cc} 1 & 0 \\ [2pt] 1 & 1 \end{array}\right]$$

• Two eigenvalues equal to 1
• Only one eigenvector, $[0,1]^T$
• Jordan block of order 2 corresponding to the repeated eigenvalue $\lambda=1$

By direct computation,

$$\mathbf{A}_2^n = \left[\begin{array}{cc} 1 & 0 \\ [2pt] n & 1 \end{array}\right]$$

which is clearly not lossless.