# dyadically recursive matrices: Part I

Introduce the $2^{n-1}\times 2^{n-1}$ matrix $B_n$ recursively as follows: $B_1(b_1)=\begin{pmatrix} b_1\end{pmatrix}$ and $$B_n(b_1,\dots,b_n)=\begin{pmatrix} B_{n-1}(b_1,\dots,b_{n-1})& b_nJ_{n-1}\\ b_nJ_{n-1}&B_{n-1}(b_1,\dots,b_{n-1}) \end{pmatrix}.$$ Here $J_n$ is a $2^{n-1}\times 2^{n-1}$ matrix with $1$'s on the antidiagonal and zeros elsewhere.

Example. For $n=2$ and $n=3$, we have $$B_2(b_1,b_2)=\begin{pmatrix} b_1&b_2\\b_2&b_1\end{pmatrix} \qquad\text{and} \qquad B_3(b_1,b_2,b_3)=\begin{pmatrix} b_1&b_2&0&b_3\\b_2&b_1&b_3&0\\0&b_3&b_1&b_2\\b_3&0&b_2&b_1\end{pmatrix}.$$

Question. Is there a closed (or interesting) formula for the determinant $\det(B_n)$?

• why is it 'dyadic'? – T.... Mar 16 '17 at 2:56
• Running in powers of $2$. – T. Amdeberhan Mar 16 '17 at 2:57
• By "dimension" of a (square) matrix, you mean the number of rows? – Gerry Myerson Mar 16 '17 at 3:43
• Can't you use Schur complement for this? The result should be an iteration of B^2 - C^2. I get $(A^2 - B^2)^2 - (C^2)^2$ for the next iterate. Gerhard "Capitalizes On Change Of Notation " Paseman, 2017.03.15. – Gerhard Paseman Mar 16 '17 at 3:52
• @GerryMyerson: Yes, the matrix is $2^{n-1}\times 2^{n-1}$. – T. Amdeberhan Mar 16 '17 at 3:58

For a vector $v$ of length $2^{n-2}$, denote by $v'$ the "mirrored" vector $J_{n-1}v$. If $v$ is an eigenvector of $B_{n-1}$ for the eigenvalue $\lambda$, then $B_n\binom v{\pm v'}=(\lambda\pm b_n)\binom v{\pm v'}$. So this gives you all the eigenvalues of $B_n$ as the sums $b_1\pm\cdots\pm b_n$ (note that $b_1$ never has a "$-$" sign) and the eigenvectors as a certain subset (respecting the dyadically recursive structure) of the set of all vectors $(1,\pm1,...,\pm 1)$ where an even number of minus signs is used.
The determinant is: $\det(B_n)=\prod(b_1\pm b_2\pm\cdots\pm b_n)$.
• Interesting to note that the determinant $\det B_n=\prod(b_1\pm\cdots\pm b_n)$ is symmetric in $b_2,...,b_n$, which is not straightforward from the recursions. – Wolfgang Mar 16 '17 at 9:03
Yes and No. On the one hand, you can prove by a simple induction that $B_n$ commutes with $J_n$. Now, $B_n$ appears to be given blockwise, where all the blocks commute with each other. This allows you to write that the determinant of $B_n$ is the determinant of determinants'' (this is false for non-commuting blocks): $$\det B_n=(\det B_{n-1})^2-(\det b_nJ_{n-1})^2=(\det B_{n-1})^2-b_n^{2^{n-1}}.$$ On the other hand, it seems difficult to exploit this relation to give a closed form for $\det B_n$. Recall that even the iteration $z\mapsto z^2-a$ has outstanding complexity, which yields Julia and Fatou sets.