Reduced Biquaternion Spectral Theory

Introduction

The reduced biquaternion algebra $\mathbb{B}_{\mathrm{r}}\cong\mathbb{C}\times\mathbb{C}$ is commutative, associative and unital, and it is not a division algebra; both facts simplify the spectral theory and separate it from the two non-commutative members of the family. In this article spectrum without qualification means the spectrum of the linear map of left multiplication, computed as a complex spectrum by complexifying that real operator; because the algebra is commutative and the coefficients are complex, the intrinsic spectrum with scalars in $\mathbb{C}$ and the operator spectrum agree, and there is no separate quaternionic spectrum as in the split biquaternion case.

The treatment is mathematically honest: every claim is either proved or stated as a definition. The algebra, its idempotents and the decomposition $\breve Q=Q_{+}\Pi_++Q_{-}\Pi_-$ are assumed from Ideals and Peirce Decomposition of the Algebra of the Reduced Biquaternions; the norm and the invertibility criterion from Norm and Invertibility in the Algebra of the Reduced Biquaternions; the elementary functions as needed from Reduced Biquaternion Elementary Functions. Throughout $\breve Q=Q_{\blacktriangle}+jQ_{\blacktriangledown}$ with $Q_{\blacktriangle}=q_{\blacktriangle}+iq'_{\blacktriangle}$, $Q_{\blacktriangledown}=q_{\blacktriangledown}+iq'_{\blacktriangledown}$, the idempotent coordinates are $Q_{+}=Q_{\blacktriangle}+Q_{\blacktriangledown}$ and $Q_{-}=Q_{\blacktriangle}-Q_{\blacktriangledown}$, and the multiplication is componentwise.

The Spectrum of an Element

Definition. The spectrum of $\breve Q$ is the set of complex $\lambda$ for which $\lambda L_{1}-L_{\breve Q}$ is not invertible, $L_{\breve Q}$ being the operator of left multiplication on the complex two-dimensional space $\mathbb{B}_{\mathrm{r}}$ with basis $1,j$. The element is regular as an element of the algebra when it is a unit, and singular otherwise.

Theorem (the spectrum is the coordinate pair). The characteristic polynomial of $L_{\breve Q}$ is $$ \boxed{\; \chi_{\breve Q}(\lambda)=\det(\lambda L_{1}-L_{\breve Q})=(\lambda-Q_{+})(\lambda-Q_{-})=\lambda^{2}-\operatorname{tr}(\breve Q)\,\lambda+\det(\breve Q) \;}, $$ so $$ \boxed{\; \sigma(\breve Q)=\{Q_{+},Q_{-}\} \;}, $$ with the two values counted with multiplicity, where $$ \operatorname{tr}(\breve Q)=Q_{+}+Q_{-}=2Q_{\blacktriangle},\qquad \det(\breve Q)=\langle \breve Q\rangle_{\mathrm{ts}}=Q_{+}Q_{-}=Q_{\blacktriangle}^{2}-Q_{\blacktriangledown}^{2} $$ are the reduced trace and the reduced determinant. The element is singular exactly when $0\in\sigma(\breve Q)$, that is when $Q_{+}=0$ or $Q_{-}=0$, which is the zero-divisor condition $\langle \breve Q\rangle_{\mathrm{ts}}=0$.

Proof. In the basis $1,j$ the matrix of $L_{\breve Q}$ is $\begin{smallmatrix}Q_{\blacktriangle}&Q_{\blacktriangledown}\\ Q_{\blacktriangledown}&Q_{\blacktriangle}\end{smallmatrix}$, whose characteristic polynomial is $(\lambda-Q_{\blacktriangle})^{2}-Q_{\blacktriangledown}^{2}=(\lambda-Q_{\blacktriangle}-Q_{\blacktriangledown})(\lambda-Q_{\blacktriangle}+Q_{\blacktriangledown})=(\lambda-Q_{+})(\lambda-Q_{-})$; the trace is $2Q_{\blacktriangle}$ and the determinant $Q_{\blacktriangle}^{2}-Q_{\blacktriangledown}^{2}=\langle \breve Q\rangle_{\mathrm{ts}}=Q_{+}Q_{-}$, and the algebra is commutative, so left and right multiplications coincide.

Corollary (the diagonal model). In the idempotent basis $\Pi_+,\Pi_-$ the operator $L_{\breve Q}$ is diagonal, $$ L_{\breve Q}=\operatorname{diag}(Q_{+},Q_{-}), $$ so the spectrum is read directly on the two coordinates and no triangular or Jordan structure appears.

Proof. $\breve Q\Pi_+=Q_{+}\Pi_+$ and $\breve Q\Pi_-=Q_{-}\Pi_-$, because the idempotents are orthogonal and $Q_{+},Q_{-}$ are the two characters.

Left and Right Eigenvalues

Theorem (the two spectra coincide). For every $\breve Q$, a vector $w$ is a left eigenvector with eigenvalue $\lambda$ if and only if it is a right eigenvector with the same eigenvalue; the left and the right spectra are equal to the spectrum $\sigma(\breve Q)$ of the preceding section.

Proof. The algebra is commutative, so $\breve Q w=\lambda w$ and $w\breve Q=\lambda w$ are the same equation; and the right spectrum is the spectrum of right multiplication, whose matrix is the transpose of $\begin{smallmatrix}Q_{\blacktriangle}&Q_{\blacktriangledown}\\ Q_{\blacktriangledown}&Q_{\blacktriangle}\end{smallmatrix}$, with the same characteristic polynomial.

Corollary (no asymmetry). The asymmetry between the left and the right spectrum of the split biquaternion algebra, and the split between the left spectrum and the S-spectrum there, do not occur: the reduced biquaternion element has one spectrum, the coordinate pair, and it is the same for the left and the right action.

Proof. Commutativity makes the two actions identical, and the conjugate classes that distinguish the quaternionic spectra degenerate to single points in $\mathbb{C}$.

Cayley–Hamilton and the Trace and Determinant Functionals

Theorem (the reduced Cayley–Hamilton identity). Every element satisfies $$ \boxed{\; \breve Q^{2}-\operatorname{tr}(\breve Q)\,\breve Q+\det(\breve Q)\,1=0,\qquad \operatorname{tr}(\breve Q)=2Q_{\blacktriangle},\quad \det(\breve Q)=\langle \breve Q\rangle_{\mathrm{ts}} \;}, $$ and the same identity holds for the operator, $L_{\breve Q}^{2}-2Q_{\blacktriangle}\,L_{\breve Q}+\langle \breve Q\rangle_{\mathrm{ts}}\,L_{1}=0$.

Proof. In the idempotent coordinates the left side is $Q_{+}^{2}-(Q_{+}+Q_{-})Q_{+}+Q_{+}Q_{-}$ in the first component and $Q_{-}^{2}-(Q_{+}+Q_{-})Q_{-}+Q_{+}Q_{-}$ in the second, both zero; the operator statement follows by applying the identity to the regular module.

Theorem (additivity and multiplicativity). The reduced trace is $\mathbb{C}$-linear and additive, $\operatorname{tr}(\breve Q_{1}+\breve Q_{2})=\operatorname{tr}(\breve Q_{1})+\operatorname{tr}(\breve Q_{2})$; the reduced determinant is multiplicative, $\det(\breve Q_{1}\breve Q_{2})=\det(\breve Q_{1})\det(\breve Q_{2})$; and $\det(\breve Q)$ vanishes exactly on the zero-divisor set.

Proof. $\operatorname{tr}(\breve Q)=2Q_{\blacktriangle}$ is linear in the complex coordinate $Q_{\blacktriangle}$; $\det(\breve Q)=\langle \breve Q\rangle_{\mathrm{ts}}=Q_{+}Q_{-}$ is multiplicative, the coordinates multiplying componentwise; and the vanishing of the product is the unit criterion.

Corollary (the inverse). On the complement of the zero-divisor set, $$ \breve Q^{-1}=Q_{+}^{-1}\Pi_++Q_{-}^{-1}\Pi_-=\frac{Q_{\blacktriangle}-jQ_{\blacktriangledown}}{\langle \breve Q\rangle_{\mathrm{ts}}}, $$ the second expression being the companion formula in the basis $1,j$.

Proof. The coordinate inverse $Q_{+}^{-1}\Pi_++Q_{-}^{-1}\Pi_-$ multiplies $\breve Q=Q_{+}\Pi_++Q_{-}\Pi_-$ to $\Pi_++\Pi_-=1$; and $Q_{\blacktriangle}-jQ_{\blacktriangledown}=(Q_{\blacktriangle}-Q_{\blacktriangledown})\Pi_++(Q_{\blacktriangle}+Q_{\blacktriangledown})\Pi_-=Q_{-}\Pi_++Q_{+}\Pi_-$, whose product with $Q_{+}^{-1}\Pi_++Q_{-}^{-1}\Pi_-$ is again $1$.

Similarity and the Classification

Definition. Two elements are similar, written $\breve Q_{1}\sim \breve Q_{2}$, when $P\breve Q_{1}P^{-1}=\breve Q_{2}$ for some unit $P$.

Theorem (the classification). $\breve Q_{1}\sim \breve Q_{2}$ if and only if $Q_{1+}=Q_{2+}$ and $Q_{1-}=Q_{2-}$, that is if and only if the two elements have the same ordered pair of idempotent coordinates; the reduced trace and the reduced determinant alone fix only the unordered pair. The similarity classes are therefore the ordered pairs $(Q_{+},Q_{-})\in\mathbb{C}^{2}$, and there is no Jordan pathology: every element is diagonal, and the only nilpotent element is $0$.

Proof. A unit $P$ has coordinates $(p,q)$ with $p,q\neq0$ and conjugates $\breve Q_{1}$ to $(pQ_{1+}p^{-1},qQ_{1-}q^{-1})=(Q_{1+},Q_{1-})$, so similarity does not change the coordinates; conversely, if the coordinates agree, the identity is a conjugating unit. No permutation of the two coordinates is induced by a unit, since the unit group acts diagonally, so the pair is ordered. Finally $\breve Q^{n}=Q_{+}^{n}\Pi_++Q_{-}^{n}\Pi_-$, which vanishes for some $n$ exactly when $Q_{+}=Q_{-}=0$.

Corollary (the commutant). The centraliser of a non-scalar element $\breve Q$, that is the set of $\breve Q_{2}$ with $\breve Q \breve Q_{2}=\breve Q_{2}\breve Q$, is the whole algebra, since the algebra is commutative; and the automorphisms of the algebra act on the coordinates by the permutations and the coordinatewise conjugations rather than by an inner action.

Proof. Commutativity makes every element commute with every other; the automorphisms act on the pair $(Q_{+},Q_{-})$ by the swaps and conjugations induced by the bar, the tessarine sign and the star, described in The Two Involutions and the Group of Involutions.

Eigenspaces and Their Dimensions

Theorem (the eigenspaces). If $Q_{+}\neq Q_{-}$, the eigenvalue $Q_{+}$ has the one-dimensional eigenspace $\mathbb{C}\Pi_+$ and the eigenvalue $Q_{-}$ the one-dimensional eigenspace $\mathbb{C}\Pi_-$, each of complex dimension one, and the two span the algebra. If $Q_{+}=Q_{-}$, that is if $Q_{\blacktriangledown}=0$ and $\breve Q=Q_{\blacktriangle}\cdot1$, then $\breve Q$ is the scalar $Q_{\blacktriangle}$, the spectrum is the single value $Q_{\blacktriangle}$ with multiplicity two, and the eigenspace is the whole algebra.

Proof. $\breve Q\Pi_+=Q_{+}\Pi_+$ and $\breve Q\Pi_-=Q_{-}\Pi_-$, and the two idempotent lines are the coordinate axes; when $Q_{+}=Q_{-}$ the operator $L_{\breve Q}$ is the scalar $Q_{\blacktriangle}$ and every vector is an eigenvector.

Corollary (the spectral projections). The spectral projection onto the eigenspace of $Q_{+}$ is $\Pi_+$ and onto that of $Q_{-}$ is $\Pi_-$, and they are the idempotents of the algebra. The minimal polynomial of $\breve Q$ is $(\lambda-Q_{+})(\lambda-Q_{-})$ when $Q_{+}\neq Q_{-}$ and $\lambda-Q_{+}$ when $Q_{+}=Q_{-}$, that is the product of the distinct linear factors of the characteristic polynomial.

Proof. The projections are the coordinate projections of the Peirce decomposition, and the minimal polynomial is the product of the distinct linear factors of the characteristic polynomial.

The Resolvent

Definition. For $\lambda\notin\sigma(\breve Q)$ the resolvent is $R(\lambda,\breve Q)=(\lambda L_{1}-L_{\breve Q})^{-1}$, read on the algebra as $(\lambda-\breve Q)^{-1}$.

Theorem (the closed form and the identity). The resolvent is $$ R(\lambda,\breve Q)=\frac{\Pi_+}{\lambda-Q_{+}}+\frac{\Pi_-}{\lambda-Q_{-}}, $$ it is analytic in $\lambda$ on the complement of the two points $Q_{+},Q_{-}$, and it satisfies the resolvent identity $R(\lambda,\breve Q)-R(\mu,\breve Q)=(\mu-\lambda)R(\lambda,\breve Q)R(\mu,\breve Q)$.

Proof. In the idempotent basis the operator is diagonal with diagonal entries $\lambda-Q_{+}$ and $\lambda-Q_{-}$, and the inverse is diagonal with the reciprocal entries; the identity is algebra in the commutative algebra.

Corollary (the resolvent at infinity). For large $\lambda$ one has $R(\lambda,\breve Q)=\lambda^{-1}1+\lambda^{-2}\breve Q+\lambda^{-3}\breve Q^{2}+\cdots$, the Neumann series of the algebra, convergent for $|\lambda|>\rho(\breve Q)$.

Proof. Expanding $(\lambda-\breve Q)^{-1}=\lambda^{-1}(1-\breve Q/\lambda)^{-1}$ and summing the geometric series in the finite-dimensional commutative algebra gives the displayed series, which converges for $|\lambda|>\rho(\breve Q)$ and sums to the closed form $(\lambda-Q_{+})^{-1}\Pi_++(\lambda-Q_{-})^{-1}\Pi_-$; the terms satisfy the recurrence $\breve Q^{n+1}=\operatorname{tr}(\breve Q)\breve Q^{n}-\det(\breve Q)\breve Q^{n-1}$ read from Cayley–Hamilton, so the series is that of the two geometric series in the coordinates.

The Spectral Radius

Definition. The spectral radius of $\breve Q$ is $\rho(\breve Q)=\max\{|\lambda|:\lambda\in\sigma(\breve Q)\}$.

Theorem (the value and the operator norm). $\rho(\breve Q)=\max(\lvert Q_{+}\rvert,\lvert Q_{-}\rvert)$; it equals the operator norm of $L_{\breve Q}$ for the Euclidean norm of the algebra, $\rho(\breve Q)=\lVert L_{\breve Q}\rVert$; it vanishes exactly at $\breve Q=0$; and it is bounded by the two coordinates through $$ \frac{\lvert Q_{+}\rvert^{2}+\lvert Q_{-}\rvert^{2}}{4}\le\rho(\breve Q)^{2}\le\lvert Q_{+}\rvert^{2}+\lvert Q_{-}\rvert^{2}. $$

Proof. The operator is diagonal with entries $Q_{+},Q_{-}$, so its norm and its spectral radius are both the maximum of the two moduli; the vanishing of the maximum forces both coordinates to zero, hence $\breve Q=0$, the algebra having no non-zero nilpotent. The left inequality compares the maximum with twice the mean and the right compares it with the sum of squares, and $\lvert Q_{+}\rvert^{2}+\lvert Q_{-}\rvert^{2}=2K(\breve Q)$.

Corollary (the Gelfand formula is finite). Because the powers are $\breve Q^{n}=Q_{+}^{n}\Pi_++Q_{-}^{n}\Pi_-$ and obey the two-term recurrence $\breve Q^{n+2}=\operatorname{tr}(\breve Q)\,\breve Q^{n+1}-\det(\breve Q)\,\breve Q^{n}$ of Cayley–Hamilton, the Gelfand limit is the maximum of the two moduli, $$ \rho(\breve Q)=\lim_{n}\lVert \breve Q^{n}\rVert^{1/n}=\max\bigl(\lvert Q_{+}\rvert,\lvert Q_{-}\rvert\bigr), $$ computed from the two coordinates alone, and there is no nilpotent block.

Proof. Cayley–Hamilton gives the recurrence for the powers, whence the closed form $\breve Q^{n}=Q_{+}^{n}\Pi_++Q_{-}^{n}\Pi_-$; with $\lVert \breve Q^{n}\rVert=\sqrt{\tfrac12(\lvert Q_{+}\rvert^{2n}+\lvert Q_{-}\rvert^{2n})}$ the $n$-th root tends to $\max(\lvert Q_{+}\rvert,\lvert Q_{-}\rvert)$ as $n\to\infty$, the larger of the two powers dominating.

The Exponential and the Logarithm

Theorem (the exponential). The exponential is componentwise, $$ \exp \breve Q=e^{Q_{+}}\Pi_++e^{Q_{-}}\Pi_-, $$ it maps the algebra into the unit group, and the reduced determinant is $\det(\exp \breve Q)=e^{Q_{+}+Q_{-}}=e^{\operatorname{tr}(\breve Q)}$. The exponential is never zero, so every element of the image is a unit; it is not injective, its kernel being the elements with $Q_{+},Q_{-}\in2\pi i\mathbb{Z}$.

Proof. The exponential of a diagonal operator is diagonal with the exponentials of the entries, and the coordinates are the characters, so $\exp(\breve Q)$ has coordinates $e^{Q_{+}},e^{Q_{-}}$; the determinant is the product, $e^{Q_{+}}e^{Q_{-}}=e^{Q_{+}+Q_{-}}$; the kernel is the preimage of $1$, that is $e^{Q_{+}}=e^{Q_{-}}=1$.

Theorem (the logarithm). An element $\breve Q$ has a logarithm, that is a solution of $\exp \breve Q_{2}=\breve Q$, if and only if it is a unit, and then $$ \log \breve Q=\log Q_{+}\,\Pi_++\log Q_{-}\,\Pi_- $$ for a choice of the two complex branches; the solution is determined modulo the kernel, which is $2\pi i\,\mathbb{Z}^{2}$ read in the two coordinates.

Proof. Each coordinate equation $e^{s}=Q_{+}$ has a solution exactly when $Q_{+}\neq0$, and the solutions differ by $2\pi i$; the two coordinates decouple.

The Relation to the Biquaternion and Split-Biquaternion Spectral Theories

The biquaternion algebra $\mathbb{B}=M_{2}(\mathbb{C})$ is simple, with a single complex eigenvalue of multiplicity two for a generic element and the S-spectrum to repair the failure of the naive definition. The split biquaternion algebra $\mathbb{H}_{\mathbb{D}}\cong\mathbb{H}\oplus\mathbb{H}$ has a four-value complex spectrum $\{q_{0}^{+}\pm i|\vec Q_{+}|\}\cup\{q_{0}^{-}\pm i|\vec Q_{-}|\}$ with doubled multiplicities, a quaternionic left spectrum $\{\tilde Q_{+},\tilde Q_{-}\}$ and a right spectrum that is a union of two conjugacy classes. The reduced biquaternion algebra is the commutative case: the spectrum is the two complex coordinates $Q_{+},Q_{-}$ with multiplicity one, the left and the right spectra coincide, the reduced trace and determinant are $\mathbb{C}$-valued and are the ordinary trace and determinant of the diagonal model, the Cayley–Hamilton identity is the ordinary one, and the resolvent is the pair of one-variable resolvents. What is lost against the quaternion cases is the conjugate-class structure; what is gained is that the two coordinates classify the element completely, with no Jordan blocks and no nilpotents beyond $0$.

Summary

The reduced biquaternion element $\breve Q$ has the spectrum $\sigma(\breve Q)=\{Q_{+},Q_{-}\}$, the pair of its idempotent coordinates, which are the eigenvalues of left multiplication in the basis $1,j$ with matrix $\begin{smallmatrix}Q_{\blacktriangle}&Q_{\blacktriangledown}\\ Q_{\blacktriangledown}&Q_{\blacktriangle}\end{smallmatrix}$. Left and right eigenvalues coincide, because the algebra is commutative, and the intrinsic spectrum over $\mathbb{C}$ is the operator spectrum; there is no quaternionic or S-spectrum. The reduced trace $\operatorname{tr}(\breve Q)=2Q_{\blacktriangle}=Q_{+}+Q_{-}$ is additive and the reduced determinant $\det(\breve Q)=\langle \breve Q\rangle_{\mathrm{ts}}=Q_{+}Q_{-}$ is multiplicative and vanishes exactly on the zero-divisor set, and Cayley–Hamilton reads $\breve Q^{2}-\operatorname{tr}(\breve Q)\breve Q+\det(\breve Q)1=0$, the minimal polynomial having degree at most two. Similarity changes neither coordinate, so the classes are the ordered pairs $(Q_{+},Q_{-})$; the eigenspaces are the two idempotent lines when $Q_{+}\neq Q_{-}$ and the whole algebra when $\breve Q$ is a scalar; the resolvent is $(\lambda-Q_{+})^{-1}\Pi_++(\lambda-Q_{-})^{-1}\Pi_-$, analytic off the two points and satisfying the resolvent identity; and the spectral radius is $\max(|Q_{+}|,|Q_{-}|)$, equal to the operator norm of left multiplication and vanishing only at $0$, there being no non-zero nilpotent. The exponential is $e^{Q_{+}}\Pi_++e^{Q_{-}}\Pi_-$, its determinant is $e^{\operatorname{tr}(\breve Q)}$, and the logarithm exists exactly on the units for a branch choice in each coordinate. Against the biquaternion and split biquaternion theories the finite complex spectrum, the field-valued trace and determinant and the diagonal model replace the S-spectrum, the quaternionic trace and norm functionals and the conjugate-class structure, the whole theory being the pair of the complex one-variable spectral theories of the two coordinates.

Summary of Notation

Symbol	Meaning
$L_{\breve Q}$	Operator of left multiplication by $\breve Q$
$Q_{+}=Q_{\blacktriangle}+Q_{\blacktriangledown}$, $Q_{-}=Q_{\blacktriangle}-Q_{\blacktriangledown}$	Idempotent coordinates, the eigenvalues
$\sigma(\breve Q)=\{Q_{+},Q_{-}\}$	Spectrum
$\operatorname{tr}(\breve Q)=2Q_{\blacktriangle}=Q_{+}+Q_{-}$	Reduced trace
$\det(\breve Q)=\langle \breve Q\rangle_{\mathrm{ts}}=Q_{+}Q_{-}=Q_{\blacktriangle}^{2}-Q_{\blacktriangledown}^{2}$	Reduced determinant
$\chi_{\breve Q}(\lambda)=\lambda^{2}-\operatorname{tr}(\breve Q)\lambda+\det(\breve Q)$	Characteristic polynomial
$R(\lambda,\breve Q)=(\lambda-Q_{+})^{-1}\Pi_++(\lambda-Q_{-})^{-1}\Pi_-$	Resolvent
$\rho(\breve Q)=\max(\lvert Q_{+}\rvert,\lvert Q_{-}\rvert)$	Spectral radius
$\exp \breve Q=e^{Q_{+}}\Pi_++e^{Q_{-}}\Pi_-$	Exponential
$\log \breve Q=\log Q_{+}\,\Pi_++\log Q_{-}\,\Pi_-$	Logarithm on the units