The Matrix Element Representation and the Biquaternion Dynamics
Introduction
The biquaternion algebra is isomorphic to the algebra of two-by-two complex matrices, and the isomorphism transports the quadratic family to the matrix family $M\mapsto M^2+N$ with $N=\mathsf{M}_2(\tilde Q_3)$, the conjugations to the operations of adjugate, conjugate transpose and their composition, and the two norms $\langle\cdot\rangle_{\natural}$ and $\|\cdot\|_E$ to the determinant and, up to the factor $\sqrt2$, the Frobenius norm. The model is the working tool of the category: every computation in the biquaternion dynamics is a computation in $M_2(\mathbb{C})$, and every invariant of the element is an invariant of its matrix. The article states what the model carries, what it does not, and where the passage has a twist.
The isomorphism and its basic properties, the trace and the determinant, the spectrum and the Cayley–Hamilton identity are Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions; the four conjugations in matrix form are in the second of these; the quadratic family is The Biquaternion Quadratic Map and Its Julia Sets; the norms are Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra. The several-variable complex theory the matrix family belongs to is Several Complex Variables and the holomorphic dynamics of Part III.
The article owns the transport of the family, the norm comparison that makes the escape notions agree, the eigenvalue and determinant invariants along the orbit, the reading of the dynamics as two-variable matrix iteration, and the warning that the isomorphism concerns the elements and not the operators on them. It does not re-derive the isomorphism or the conjugations.
Standing convention. $\mathsf{M}_2:\mathbb{B}\to M_2(\mathbb{C})$ is the matrix realization, $M=\mathsf{M}_2(\tilde Q_1)$, $N=\mathsf{M}_2(\tilde Q_3)$, and the matrix family is
$$ \mathsf{M}_2\circ F_{\tilde Q_3}\circ\mathsf{M}_2^{-1}=\bigl(M\mapsto M^2+N\bigr). $$
The model of an element is the matrix
$$ \mathsf{M}_2(\tilde Q_1)=\begin{pmatrix}Q_{1t}-iQ_{1z}&-iQ_{1x}-Q_{1y}\\-iQ_{1x}+Q_{1y}&Q_{1t}+iQ_{1z}\end{pmatrix},\qquad Q_{1,\mu}=q_\mu+iq'_\mu , $$
of determinant $\langle\tilde Q_1\rangle_{\natural}=Q_{1t}^2+Q_{1x}^2+Q_{1y}^2+Q_{1z}^2$ and trace $2Q_{1t}$.
The Model as a Model of the Dynamics
Theorem (transport of the family). For every $n$, $\mathsf{M}_2(F_{\tilde Q_3}^n(\tilde Q_1))=p_n(\mathsf{M}_2(\tilde Q_1))$, where $p_0(M)=M$ and $p_{n+1}(M)=p_n(M)^2+N$. Consequently the map $\mathsf{M}_2$ restricts to a bijection
$$ \boxed{\;\mathcal K_{\tilde Q_3}\;\longleftrightarrow\;\{M\in M_2(\mathbb{C}) : (p_n(M))_{n\ge0} \text{ bounded}\}, \qquad J_{\tilde Q_3}\leftrightarrow \text{the boundary}\;}, $$
and the dynamics of the biquaternion family is conjugate to the dynamics of a polynomial map of the four complex entries of the matrix.
Proof. $\mathsf{M}_2$ is multiplicative and $\mathbb{C}$-linear, so $\mathsf{M}_2(\tilde Q_1^2+\tilde Q_3)=\mathsf{M}_2(\tilde Q_1)^2+\mathsf{M}_2(\tilde Q_3)$ and induction gives the transport of the orbit; a bijective linear map preserves boundedness and boundaries.
Remark (the reduction to four complex coordinates). The matrix has four complex entries; with $M=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}$ its square is
$$ M^2=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix}\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix} =\begin{pmatrix}\alpha^2+\beta\gamma&\beta(\alpha+\delta)\\\gamma(\alpha+\delta)&\delta^2+\beta\gamma\end{pmatrix}, $$
and the family is the four-dimensional complex map
$$ (\alpha,\beta,\gamma,\delta)\longmapsto(\alpha^2+\beta\gamma+n_1,\ \beta(\alpha+\delta)+n_2,\ \gamma(\alpha+\delta)+n_3,\ \gamma\beta+\delta^2+n_4) $$
with $N=\begin{pmatrix}n_1&n_2\\n_3&n_4\end{pmatrix}$; the diagonal entries are quadratic in one line and the off-diagonal entries are linear in the trace $\alpha+\delta$ and the off-diagonal pair. The map of the entries is a polynomial map of degree two of $\mathbb{C}^4$, and the four complex dimensions of the space appear here for the first time.
The Norm Comparison and the Escape Notions
Proposition (the Frobenius and Euclidean norms). For every biquaternion, $\|\mathsf{M}_2(\tilde Q_1)\|_F=\sqrt2\,\|\tilde Q_1\|_E$; for every pair, $\|\tilde Q_2\tilde Q_1\|_E\le\sqrt2\,\|\tilde Q_2\|_E\|\tilde Q_1\|_E$; and the induced operator norm satisfies $\|M\|_2\le\sqrt2\,\|\tilde Q_1\|_E$.
Proof. The Frobenius norm of the explicit matrix $\mathsf{M}_2(\tilde Q_1)$ displayed above is the squared sum of the moduli of the entries, which is $\sum_\mu|Q_{1,\mu}|^2$ computed twice, hence $\|\mathsf{M}_2(\tilde Q_1)\|_F^2=2\|\tilde Q_1\|_E^2$. Multiplicativity of $\mathsf{M}_2$ and submultiplicativity of the Frobenius norm give $\sqrt2\|\tilde Q_2\tilde Q_1\|_E=\|\mathsf{M}_2(\tilde Q_2\tilde Q_1)\|_F\le\|\mathsf{M}_2(\tilde Q_2)\|_F\|\mathsf{M}_2(\tilde Q_1)\|_F=2\|\tilde Q_2\|_E\|\tilde Q_1\|_E$, hence the second inequality. The operator norm is bounded by the Frobenius norm, which is $\sqrt2\|\tilde Q_1\|_E$.
Corollary (the escape notions agree). A biquaternion orbit is bounded in the Euclidean norm exactly when its matrix orbit is bounded in the Frobenius norm, and exactly when it is bounded in the operator norm. The filled Julia set and the Julia set are the same sets in the two models.
Proof. The three norms are pairwise comparable on the fixed finite-dimensional space, by the proposition, and a sequence is bounded for one exactly when it is bounded for the others. The biquaternion norm $\langle\cdot\rangle_{\natural}$ is a different object: it is the determinant of the matrix and the product $\tilde Q_1\tilde Q_1^{\natural}$, it is complex-valued and indefinite and vanishes on the zero divisors, so boundedness of $\langle\cdot\rangle_{\natural}$ is not equivalent to boundedness in the Euclidean norm — a nilpotent element has $N=0$ and arbitrarily large Euclidean norm.
Remark (the constant $\sqrt2$ is the only discrepancy). Every statement of the category phrased in a norm differs between the algebra and the model by at most the factor $\sqrt2$, and the escape radius of the model and of the algebra differ accordingly. The factor is a normalisation and not a phenomenon, but it must be tracked, because a radius of $2$ in the model is a radius of $\sqrt2$ in the algebra.
The Invariants Along the Orbit
Proposition (determinant and trace). Along the orbit,
$$ \boxed{\;\det\mathsf{M}_2(\tilde Q_{1,n})=\langle\tilde Q_{1,n}\rangle_{\natural}, \qquad \operatorname{tr}\mathsf{M}_2(\tilde Q_{1,n})=2(Q_{1,n})_0\;}, $$
and for the parameter zero, $\langle\tilde Q_{1,n}\rangle_{\natural}=\langle\tilde Q_1\rangle_{\natural}^{2^n}$. In the entries of the matrix the two invariants are the classical ones,
$$ M=\begin{pmatrix}\alpha&\beta\\\gamma&\delta\end{pmatrix},\qquad \det M=\alpha\delta-\beta\gamma,\qquad \operatorname{tr}M=\alpha+\delta,\qquad \chi_M(\zeta)=\zeta^2-(\operatorname{tr}M)\zeta+(\det M). $$
Proof. $\det\mathsf{M}_2=N$ and $\operatorname{tr}\mathsf{M}_2=2Q_{1t}$ are the trace-and-determinant proposition of the matrix-element-representation article, applied to the iterate. For a central zero parameter $\mathsf{M}_2(\tilde Q_{1,n})=\mathsf{M}_2(\tilde Q_1)^{2^n}$ and the determinant is multiplicative.
Theorem (the spectrum and the central parameter). The spectrum of $\mathsf{M}_2(\tilde Q_{1,n})$ is $\{q_n(\lambda_1),q_n(\lambda_2)\}$ for the spectrum $\{\lambda_1,\lambda_2\}$ of $\mathsf{M}_2(\tilde Q_1)$, where $q_n$ are the iterates of $\zeta\mapsto\zeta^2+C$ when $\tilde Q_3=Ce_t$; in particular the trace and the determinant of the iterate are the elementary symmetric functions of the two transported eigenvalues.
Proof. The functional calculus of the polynomial $p_n$ on a two-by-two matrix and the elementary symmetric functions of the spectrum. This is the eigenvalue reduction of The Biquaternion Quadratic Map and Its Julia Sets, here read as a statement of the model.
The Two-Variable Interpretation
The model turns the biquaternion dynamics into the iteration of a polynomial map of $\mathbb{C}^4$ with the special form above, and the general theory of such maps is the pluripotential and the Fatou theory of several variables.
Remark (what the two-variable theory gives and what it needs). For a polynomial map of $\mathbb{C}^4$ whose leading homogeneous part is proper, there is a Green's function, an equilibrium measure, a filled Julia set and a Fatou theory, and the equilibrium measure is the harmonic measure of the fractal (Several Complex Variables, Plurisubharmonic Functions, and the pluripotential article of the category). The leading part of the matrix family is $M\mapsto M^2$, whose fibre over $0$ is the square-zero cone of the traceless singular matrices and is not finite; the several-variable theory therefore applies to the restriction to the regular part and not to the whole map. This is the model form of the obstruction of The Zero Divisors and the Singular Julia Sets.
Remark (the rank-one cone in the model). In the model the zero divisors are the non-zero singular matrices; the determinant criterion, the rank-one elements and the outer product are Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions, and the critical set is the union of the singular matrices with the traceless ones. The two components of the critical set are the two classical degenerations of a two-by-two matrix, singular and traceless, and this is the cleanest statement of the critical set in the whole category.
The Warning: Elements and Operators
The isomorphism is an isomorphism of algebras, and an algebra isomorphism carries multiplication and its derived operations; it does not carry every operator that can be built on the underlying space.
Proposition (the element and the operator are different objects). Let $L_{\tilde Q_1}:\mathbb{B}\to\mathbb{B}$ be left multiplication, $L_{\tilde Q_1}(\tilde Q_2)=\tilde Q_1\tilde Q_2$. Then $L_{\tilde Q_1}$ is a complex-linear endomorphism of the four-dimensional space, represented by a $4\times4$ matrix in any basis, while $\mathsf{M}_2(\tilde Q_1)$ is the two-by-two matrix corresponding to the element $\tilde Q_1$. In the basis $e_t,\vec e_x,\vec e_y,\vec e_z$ the matrix of $L_{\tilde Q_1}$ is
$$ \begin{pmatrix} Q_{1t}&-Q_{1x}&-Q_{1y}&-Q_{1z}\\ Q_{1x}&Q_{1t}&-Q_{1z}&Q_{1y}\\ Q_{1y}&Q_{1z}&Q_{1t}&-Q_{1x}\\ Q_{1z}&-Q_{1y}&Q_{1x}&Q_{1t} \end{pmatrix}, $$
of determinant $\langle\tilde Q_1\rangle_{\natural}^2$ and trace $4Q_{1t}$, with characteristic polynomial $\bigl((\zeta-Q_{1t})^2+\sum_kQ_k^2\bigr)^2$. The two are different objects: $\mathsf{M}_2(L_{\tilde Q_1})$ is not defined, and the eigenvalues of the $4\times4$ matrix $L_{\tilde Q_1}$ are $\{Q_{1t}+iB,Q_{1t}+iB,Q_{1t}-iB,Q_{1t}-iB\}$ with $B=\sqrt{\sum_kQ_k^2}$, the eigenvalues of $\mathsf{M}_2(\tilde Q_1)$ each counted twice.
Proof. Under the identification $\mathbb{B}=\mathbb{C}^4$ the operator $L_{\tilde Q_1}$ is the left regular representation, and its characteristic polynomial is the square of that of $\mathsf{M}_2(\tilde Q_1)$ because $\mathbb{B}\cong M_2(\mathbb{C})$ is the direct sum of two copies of the simple module on which $M=\mathsf{M}_2(\tilde Q_1)$ acts by $M$ (Modules over the Biquaternion Algebra). The eigenvalues of $M$ are $Q_{1t}\pm iB$ by the characteristic polynomial of the matrix-element-representation article.
Corollary (where the operator enters the dynamics). The derivative of the quadratic family is the sum of two operators on the space and not an element,
$$ \boxed{\;dF_{\tilde Q_3}\big|_{\tilde Q_1}(\tilde Q_2)=\tilde Q_1\tilde Q_2+\tilde Q_2\tilde Q_1=L_{\tilde Q_1}(\tilde Q_2)+R_{\tilde Q_1}(\tilde Q_2)\;}, $$
which is why the critical set is read from the pair of eigenvalues of $\mathsf{M}_2(\tilde Q_1)$ — $\det dF=4\langle\tilde Q_1\rangle_{\natural}(2Q_{1t})^2$ — and not from the matrix of the element alone.
Proof. The product rule; the determinant computation is the critical-set proposition of The Biquaternion Quadratic Map and Its Julia Sets.
Remark (the moral of the warning). A statement about the element $\tilde Q_1$ transports along $\mathsf{M}_2$ and can be read from the two-by-two matrix; a statement about the operator $\tilde Q_2\mapsto\tilde Q_1\tilde Q_2$ or about a product of the space does not, and must be proven in the algebra. The model is a model of the elements and of their multiplication, and every use of the model must name which of the two objects is meant. The four general products of the space are the standard trap: they are not carried by $\mathsf{M}_2$, which is the isomorphism for one product only.
Summary
The matrix model conjugates the biquaternion quadratic family to the polynomial map $M\mapsto M^2+N$ of the four complex entries of a two-by-two matrix, carries the filled Julia set and the Julia set bijectively to their matrix counterparts, and compares the two norms by the single factor $\sqrt2$: the Frobenius norm of the matrix is $\sqrt2$ times the Euclidean norm of the element, and the submultiplicativity constant is $\sqrt2$. The determinant and the trace of the iterate are the biquaternion norm and twice the scalar part, the spectrum of the iterate is the image of the spectrum under the scalar quadratic map for a central parameter, and the two-variable complex theory applies to the model on the regular part, its obstruction being the square-zero cone that its leading part is not proper over. The article closes with the warning that the isomorphism concerns the elements and not the operators built on them: left multiplication is a $4\times4$ operator with the eigenvalues of the element doubled, and the derivative of the family is the sum of two such operators, which is where the operator enters and the reason the critical set is the union of the singular matrices and the traceless ones.
Summary of Notation
| symbol | meaning |
|---|---|
| $\mathsf{M}_2:\mathbb{B}\to M_2(\mathbb{C})$ | the matrix realization |
| $M=\mathsf{M}_2(\tilde Q_1)$, $N=\mathsf{M}_2(\tilde Q_3)$ | the matrix of the point and of the parameter |
| $\|\cdot\|_F$, $\|\cdot\|_2$ | the Frobenius and operator norms |
| $\|\cdot\|_E$ | the Euclidean norm on $\mathbb{B}$ |
| $p_n$ | the iterated polynomial, $p_{n+1}=p_n^2+N$ |
| $L_{\tilde Q_1}$, $R_{\tilde Q_1}$ | left and right multiplication operators |
| $dF_{\tilde Q_3}=\tilde Q_1\tilde Q_2+\tilde Q_2\tilde Q_1$ | the derivative, the sum of two operators |
| $\det dF=4\langle\tilde Q_1\rangle_{\natural}(2Q_{1t})^2$ | the determinant of the derivative |
Further Reading
- Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions and Introduction to the 2×2 Matrix Element Representation $M_2(\mathbb{C})$ of Biquaternions, for the isomorphism, the trace and the determinant, the spectrum and the conjugations in matrix form.
- The Biquaternion Quadratic Map and Its Julia Sets, for the family, the eigenvalue reduction and the critical set.
- Modules over the Biquaternion Algebra, for the regular representation and the doubling of the spectrum.
- The Zero Divisors and the Singular Julia Sets, for the cone that the leading part of the matrix family is not proper over.
- Quaternionic Norm Regularity and Degeneracy in the Biquaternion Algebra, for the norm $\langle\cdot\rangle_{\natural}$ and the Euclidean norm compared here.