Paper IV: A finite algebra and matrix model

6 A finite algebra and matrix model

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The central idea

The signed shift is a finite carrier for the differential cycle. At order four, three basis steps have coefficient one and the final step has coefficient minus one. Four steps return the vector with reversed sign. The matrix exponential packages every finite path length in one analytic expression.

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A four-node cycle has three positive edges and one negative closing edge; the accompanying target-row, source-column matrix has fourth power minus identity. — The exact signed shift \(J_4\). Source columns and target rows describe the same action as the cycle on the left. Direct integer multiplication verifies \(J_4^4=-I\).

The derivative and addition laws can be read from one finite algebra identity. This formulation also identifies the exponent polygon as the spectrum of a signed cyclic shift.

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Definition 6.1

Let

\[ \mathfrak A_N^+={\mathbb C}[\xi]/(\xi^N-1) \]

and

\[ \mathfrak A_N^-={\mathbb C}[\xi]/(\xi^N+1). \]

Every element has a unique representative of degree below \(N\).

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Theorem 6.2: Finite algebra exponential identities

In \(\mathfrak A_N^+\) one has

\[ e^{\lambda z\xi} = \sum_{j=0}^{N-1}{\mathsf E}_{N,j}(\lambda z)\xi^j. \]

In \(\mathfrak A_N^-\) one has

\[ e^{z\xi} = \sum_{j=0}^{N-1}{\mathsf D}_{N,j}(z)\xi^j. \]

Differentiation and multiplication of these identities recover the cyclic derivative systems and the two addition laws.

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Proof of Theorem 6.2

Expand the exponential and write every exponent uniquely as \(Nn+j\). In \(\mathfrak A_N^+\), the relation \(\xi^{Nn+j}=\xi^j\) gives the transfer sections. In \(\mathfrak A_N^-\), the relation \(\xi^{Nn+j}=(-1)^n\xi^j\) gives the signed sections.

Differentiation multiplies by \(\lambda\xi\) in the first algebra and by \(\xi\) in the second. Reduction of the product by the defining relation gives the derivative cycles. The identity

\[ e^{(z+w)\xi}=e^{z\xi}e^{w\xi} \]

gives the addition laws after comparison of the coefficients of \(1,\xi,\ldots,\xi^{N-1}\).

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Definition 6.3

Let \(J_N\) be the signed cyclic shift on the standard basis \(e_0,\ldots,e_{N-1}\) of \({\mathbb C}^N\), defined by

\[ J_Ne_j=e_{j+1} \quad (0\leq j<N-1), \qquad J_Ne_{N-1}=-e_0. \]

Then \(J_N^N=-I\).

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Theorem 6.4: Companion matrix realization

Put

\[ \mathbf D_N(z) = \begin{pmatrix} {\mathsf D}_{N,0}(z)\\ {\mathsf D}_{N,1}(z)\\ \vdots\\ {\mathsf D}_{N,N-1}(z) \end{pmatrix}. \]

Then

\[ \mathbf D_N(z)=e^{zJ_N}e_0 \]

and

\[ {\mathsf D}_N(z)=\frac1N{\operatorname{tr}}(e^{zJ_N}). \]

The eigenvalues of \(J_N\) are the \(N\) roots of \(-1\), each with multiplicity one.

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Proof of Theorem 6.4

The vector \(e^{zJ_N}e_0\) has derivative \(J_Ne^{zJ_N}e_0\) and initial value \(e_0\). Its components therefore satisfy the same signed derivative cycle and initial data as \(\mathbf D_N\). Uniqueness for the linear system gives the first identity.

The vectors

\[ e_0,J_Ne_0,\ldots,J_N^{N-1}e_0 \]

are the standard basis. Thus \(e_0\) is cyclic for \(J_N\), so the minimal polynomial of \(J_N\) has degree \(N\). Since \(J_N^N=-I\), that minimal polynomial divides \(x^N+1\) and therefore equals \(x^N+1\). The characteristic polynomial has the same degree and is divisible by the minimal polynomial, so it too equals \(x^N+1\). This polynomial has distinct roots. Consequently \(J_N\) is diagonalizable and its spectrum consists of all \(N\) roots of \(-1\), each once. Hence

\[ {\operatorname{tr}}(e^{zJ_N}) = \sum_{\eta^N=-1}e^{\eta z} = N{\mathsf D}_N(z) \]

by Theorem 5.6.

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Corollary 6.5: Entries of the fundamental matrix

For \(0\leq p,q<N\), let \(d_{pq}\) be the member of \(\{0,\ldots,N-1\}\) congruent to \(p-q\) modulo \(N\). Put

\[ \varepsilon_{pq} = \begin{cases} 1,&p\geq q,\\ -1,&p<q. \end{cases} \]

Then

\[ \bigl(e^{zJ_N}\bigr)_{pq} = \varepsilon_{pq}{\mathsf D}_{N,d_{pq}}(z). \]

Thus every entry of the fundamental matrix is one of the signed sections, with the sign recording whether the cyclic path from \(q\) to \(p\) crosses the signed edge.

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Proof of Corollary 6.5

For every nonnegative integer \(n\),

\[ J_N^{Nn+d_{pq}}e_q = (-1)^n\varepsilon_{pq}e_p. \]

Only powers whose exponent is congruent to \(d_{pq}\) modulo \(N\) contribute to the \((p,q)\) entry of the exponential. Therefore

\[ \bigl(e^{zJ_N}\bigr)_{pq} = \varepsilon_{pq} \sum_{n\geq0}(-1)^n \frac{z^{Nn+d_{pq}}}{(Nn+d_{pq})!}. \]

This is the stated formula.

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Corollary 6.6: Addition and inversion laws for the companion matrix

Assume \(N\geq2\). The matrix

\[ U_N(z)=e^{zJ_N} \]

satisfies

\[ U_N(z+w)=U_N(z)U_N(w), \qquad U_N(z)^{-1}=U_N(-z), \qquad \det U_N(z)=1. \]

Its first column is \(\mathbf D_N(z)\), and all of its entries are given by Corollary 6.5.

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Proof of Corollary 6.6

The first two identities are the exponential group law. Since \({\operatorname{tr}} J_N=0\),

\[ \det(e^{zJ_N})=e^{z{\operatorname{tr}} J_N}=1. \]

The first column statement is Theorem 6.4.

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