Dual-Numbers Matrix Element Representation

Introduction

This article constructs the faithful two-dimensional matrix representation of the dual-number algebra $\mathbb{D}'_R = R[\varepsilon]/(\varepsilon^2)$ and identifies the dual numbers with a commutative subalgebra of the $2\times2$ matrices. It follows Dual-Numbers Algebra for the conventions, Dual-Numbers Norm and Invertibility for the norm, and Dual-Number Subspaces for the two distinguished submodules. Its structural model is Biquaternion 2×2 Matrix Element Representation, in which the biquaternion algebra is identified with the full matrix algebra $M_2(\mathbb{C})$; the dual algebra is identified only with a commutative subalgebra, and the comparison at the end records the difference.

The treatment is mathematically honest: every claim is either proved or stated as a definition. No physics is invoked. Throughout, $R$ is a commutative ring with identity in which $2$ is invertible; the geometric specialisation is $R = \mathbb{R}$, and then the algebra is written $\mathbb{D}'$. A general dual number is

$$ A = a + \varepsilon a', \qquad a, a' \in R, $$

with $a = \operatorname{Re} A$, $a' = \operatorname{Inf} A$, dual conjugation $\bar A = a - \varepsilon a'$, norm $N(A) = A\bar A = a^2$, maximal ideal $\mathrm{M} = (\varepsilon)$, real submodule $R_{\mathbb{D}'}$ and infinitesimal submodule $\varepsilon R_{\mathbb{D}'}$.

The Representation

Definition

Definition. The matrix representation of $\mathbb{D}'_R$ is the map

$$ \Phi : \mathbb{D}'_R \longrightarrow M_2(R), \qquad \Phi(a + \varepsilon a') = \begin{pmatrix} a & a' \\ 0 & a \end{pmatrix}. $$

The matrix $\Phi(A)$ is the matrix of $A$; its diagonal entry is the real part and its strictly upper-triangular entry is the infinitesimal part.

Linearity and the Identity

$\Phi$ is $R$-linear by construction, and it preserves the identity:

$$ \Phi(A + B) = \Phi(A) + \Phi(B), \qquad \Phi(\lambda A) = \lambda\,\Phi(A), \qquad \Phi(1) = I = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix}. $$

The Representation Is an Algebra Homomorphism

Multiplicativity

Theorem. $\Phi$ is a unital algebra homomorphism:

$$ \Phi(AB) = \Phi(A)\,\Phi(B) \qquad \text{for all } A, B \in \mathbb{D}'_R. $$

Proof. With $A = a + \varepsilon a'$ and $B = b + \varepsilon b',$

$$ \Phi(A)\Phi(B) = \begin{pmatrix} a & a' \\ 0 & a \end{pmatrix}\begin{pmatrix} b & b' \\ 0 & b \end{pmatrix} = \begin{pmatrix} a b & a b' + a' b \\ 0 & a b \end{pmatrix} = \Phi(AB), $$

using $AB = a b + (a b' + a' b)\varepsilon$.

Injectivity

Theorem. $\Phi$ is injective, and hence a ring isomorphism onto its image.

Proof. If $\Phi(A) = 0$ then $a = 0$ and $a' = 0$, so $A = 0$.

So $\mathbb{D}'_R$ is isomorphic to the subalgebra $\Phi(\mathbb{D}'_R) \subseteq M_2(R)$, and the representation is faithful. The algebra is thus a concrete algebra of matrices, and everything proved about $\mathbb{D}'_R$ can be read in $M_2(R)$.

The Image: The Toeplitz Subalgebra

Definition. A matrix $\begin{pmatrix} a & b \\ 0 & c \end{pmatrix}$ is upper triangular Toeplitz if its diagonal entries are equal, $a = c$.

Theorem. The image $\Phi(\mathbb{D}'_R)$ is exactly the subalgebra of upper-triangular Toeplitz matrices,

$$ \Phi(\mathbb{D}'_R) = \left\{\begin{pmatrix} a & a' \\ 0 & a \end{pmatrix} : a, a' \in R\right\}. $$

It is a commutative unital subalgebra of $M_2(R)$, of $R$-rank two, spanned by $I$ and the nilpotent $E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$.

Proof. The displayed set is the image by definition; it is closed under addition and multiplication, since

$$ \begin{pmatrix} a & a' \\ 0 & a \end{pmatrix}\begin{pmatrix} b & b' \\ 0 & b \end{pmatrix} = \begin{pmatrix} a b & a b'+a' b \\ 0 & a b \end{pmatrix}, $$

which is again upper-triangular Toeplitz; the unit is $I$; and $E^2 = 0$.

The Nilpotent

Proposition. $\Phi(\varepsilon) = E$ and every element of the image is $aI + a'E$; the algebra is the algebra of truncated polynomials in $E$ at $E^2 = 0$.

Proof. $\Phi(\varepsilon) = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = E$, and $\Phi(a + \varepsilon a') = aI + a'E$ by linearity.

The Trace and the Determinant

Theorem. For $A = a + \varepsilon a',$

$$ \operatorname{tr}\Phi(A) = 2a, \qquad \det\Phi(A) = a^2 = N(A), \qquad \chi_{\Phi(A)}(\lambda) = (\lambda - a)^2. $$

Proof. The trace is the sum of the diagonal entries $a + a = 2a$; the determinant is $a^2 - 0 = a^2$; the characteristic polynomial is the product of the diagonal entries of $\lambda I - \Phi(A)$, namely $(\lambda - a)^2$, the matrix being triangular.

Corollary. The determinant is the norm and is multiplicative, $\det\Phi(AB) = \det\Phi(A)\det\Phi(B)$, which is the multiplicativity of $N$ read in the matrix algebra; the trace is twice the real part.

Corollary. The eigenvalues of $\Phi(A)$ are both equal to $a$, the real part. The matrix is diagonalizable only when $a' = 0$, in which case it is the scalar matrix $aI$; otherwise it is a nondiagonalizable Jordan block.

Remark. The determinant vanishes if and only if the eigenvalues vanish, that is if and only if $a = 0$; so $\Phi(A)$ is invertible if and only if $a$ is a unit of $R$, recovering the criterion of Dual-Numbers Norm and Invertibility from the matrix side.

Corollary (the polar form). For $A, B \in \mathbb{D}'_R$ the polar form of the norm is read from the trace and the determinant:

$$ B(A, B) = \tfrac{1}{2}\bigl(\operatorname{tr}\Phi(A)\operatorname{tr}\Phi(B) - \operatorname{tr}(\Phi(A)\Phi(B))\bigr). $$

Proof. The identity $\det(X + Y) - \det X - \det Y = \operatorname{tr}(X)\operatorname{tr}(Y) - \operatorname{tr}(XY)$ holds for all $2\times2$ matrices; substituting $\operatorname{tr}\Phi(A) = 2a$, $\operatorname{tr}\Phi(B) = 2b$ and $\operatorname{tr}(\Phi(A)\Phi(B)) = \operatorname{tr}\Phi(AB) = 2ab$ gives $\tfrac{1}{2}(4ab - 2ab) = ab = B(A,B)$.

The Image of the Distinguished Submodules

Theorem. The two distinguished submodules map onto the scalar and the strictly upper-triangular matrices:

$$ \Phi(R_{\mathbb{D}'}) = \left\{\begin{pmatrix} a & 0 \\ 0 & a \end{pmatrix} : a \in R\right\} = R\,I, \qquad \Phi(\varepsilon R_{\mathbb{D}'}) = \left\{\begin{pmatrix} 0 & a' \\ 0 & 0 \end{pmatrix} : a' \in R\right\} = R\,E. $$

Proof. An element of $R_{\mathbb{D}'}$ is $a$, mapping to $aI$; an element of $\varepsilon R_{\mathbb{D}'}$ is $\varepsilon a'$, mapping to $a'E$.

Corollary. The matrix model carries the direct-sum decomposition to the decomposition $M_2(R) \supseteq R\,I \oplus R\,E$, with $\Phi(R_{\mathbb{D}'}) \cap \Phi(\varepsilon R_{\mathbb{D}'}) = 0$ and sum $\Phi(\mathbb{D}'_R)$.

Corollary. The maximal ideal corresponds to the nilpotent line $R\,E$, and the zero divisors of Dual-Numbers Zero Divisors correspond to the nonzero scalar multiples of $E$.

The Conjugations in Matrix Form

Theorem. Dual conjugation becomes the conjugation by the diagonal involution $J = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$:

$$ \Phi(\bar A) = J\,\Phi(A)\,J^{-1} = J\,\Phi(A)\,J = \begin{pmatrix} a & -a' \\ 0 & a \end{pmatrix}. $$

Proof. $J^2 = I$ and $J\begin{pmatrix} a & a' \\ 0 & a \end{pmatrix}J = \begin{pmatrix} a & -a' \\ 0 & a \end{pmatrix}$.

Corollary. The real submodule is the fixed subalgebra of the inner involution $\operatorname{Ad}_J$, and the infinitesimal submodule is the $(-1)$-eigenspace; in matrix form these are the scalar matrices and the strictly upper-triangular matrices.

Remark (the transpose and the regular representation). The multiplication convention of the matrix model puts the infinitesimal part in the upper-right corner. The regular representation of Dual-Numbers Element Representations and Shears and Parabolic Rotations, computed in the basis $(1, \varepsilon)$ of left multiplication, instead puts it in the lower-left corner:

$$ [u] = \begin{pmatrix} a & 0 \\ a' & a \end{pmatrix}, \qquad u = a + \varepsilon a'. $$

The two conventions are related by transpose, $[u] = \Phi(u)^{\mathsf{T}}$, and the transpose reverses the order of multiplication, so $[u][v] = \Phi(vu)^{\mathsf{T}} = \Phi(uv)^{\mathsf{T}}$ because the algebra is commutative. Both are faithful representations of $\mathbb{D}'_R$ by $2\times2$ matrices; the classification of representations in Dual-Numbers Element Representations shows that this is essentially the only faithful two-dimensional one, up to the choice of basis in which the nilpotent is a Jordan block.

Structural Consequences

Commutativity and the Center

The image is commutative, so $\Phi$ cannot be surjective onto $M_2(R)$ unless the base ring is the zero ring; the image is a proper subalgebra of dimension two inside the four-dimensional $M_2(R)$. The center of the image is the whole image, and the centralizer of a non-scalar element $aI + a'E$, $a' \neq 0$, is exactly the image; the centralizer of a scalar matrix is all of $M_2(R)$.

Units

An upper-triangular Toeplitz matrix $aI + a'E$ is invertible if and only if $a$ is a unit of $R$, and then

$$ (aI + a'E)^{-1} = a^{-1}I - a^{-2}a'\,E = \begin{pmatrix} a^{-1} & -a^{-2}a' \\ 0 & a^{-1} \end{pmatrix}, $$

matching $\Phi(A^{-1})$ for $A = a + \varepsilon a'$. The unit group of the image is the set of matrices $aI + a'E$ with $a \in R^\times$, whose shear subgroup is $I + R\,E = \{I + a'E\} \cong (R,+)$.

Automorphisms in the Matrix Model

Proposition. The automorphism $\varphi_c(\varepsilon) = \varepsilon c$ with $c \in R^\times$ of Dual-Numbers Automorphisms and Derivations is realised by conjugation with the diagonal matrix $\operatorname{diag}(c, 1)$:

$$ \operatorname{diag}(c, 1)\,\Phi(A)\,\operatorname{diag}(c, 1)^{-1} = \begin{pmatrix} a & c a' \\ 0 & a \end{pmatrix} = \Phi\bigl(\varphi_c(A)\bigr). $$

Proof. $\operatorname{diag}(c,1)^{-1} = \operatorname{diag}(c^{-1},1)$, and multiplying the three matrices gives $\begin{pmatrix} a & c a' \\ 0 & a \end{pmatrix}$.

So $\operatorname{Aut}_R(\mathbb{D}'_R)$ is the image of the diagonal subgroup of $GL_2(R)$ acting by conjugation: a diagonal element $\operatorname{diag}(c, d)$ rescales the $E$-coordinate by $c d^{-1}$, and the induced homomorphism $(c,d) \mapsto c d^{-1}$ onto $R^\times$ is surjective with kernel the scalar matrices $\operatorname{diag}(c, c)$.

Comparison with the $2\times2$ Representation of $\mathbb{H}$ and the Absence for $\mathbb{H}_{\mathbb{D}}$

The Quaternion Case

The quaternion algebra $\mathbb{H}$ has the faithful $2\times2$ representation over $\mathbb{C}$

$$ \Psi(a + b e_1 + ce_2 + de_3) = \begin{pmatrix} a + i b & c + i d \\ -c + i d & a - i b \end{pmatrix}, \qquad i = \sqrt{-1}, $$

with determinant $a^2 + b^2 + c^2 + d^2$, the non-degenerate norm of $\mathbb{H}$; the image is not a subalgebra of $M_2(\mathbb{R})$ but a real form inside $M_2(\mathbb{C})$, of real dimension four, which realifies to a four-dimensional subalgebra of $M_4(\mathbb{R})$.

The dual-number model is the analogue in which the coefficient ring stays $\mathbb{R}$ and the image is the commutative Toeplitz subalgebra; the determinant is the degenerate norm $a^2$, and the missing imaginary part of the coefficients is exactly the lost second coordinate. Both models sit inside a four-real-dimensional matrix algebra, but the quaternion image is a four-real-dimensional real form of $M_2(\mathbb{C})$, while the dual image is only the two-real-dimensional commutative subalgebra $RI \oplus RE$ of $M_2(\mathbb{R})$: $\mathbb{H}$ needs complex coefficients and $\mathbb{D}'$ does not.

The Absence of a Two-Dimensional Representation for $\mathbb{H}_{\mathbb{D}}$

The split biquaternion algebra $\mathbb{H}_{\mathbb{D}} = \mathbb{D}\otimes_{\mathbb{R}}\mathbb{H}$ has real dimension eight, and $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\oplus\mathbb{H}$. There is no faithful representation of $\mathbb{H}_{\mathbb{D}}$ by $2\times2$ matrices over $\mathbb{R}$, and none over $\mathbb{C}$. Over $\mathbb{R}$, a faithful representation would be an injective $\mathbb{R}$-linear map into $M_2(\mathbb{R})$, which would need $\dim_{\mathbb{R}} M_2(\mathbb{R}) = 4 \ge \dim_{\mathbb{R}}\mathbb{H}_{\mathbb{D}} = 8$, and this fails. Over $\mathbb{C}$ the dimension count alone does not obstruct: a homomorphism of $\mathbb{R}$-algebras $\mathbb{H}_{\mathbb{D}} \to M_2(\mathbb{C})$ is only $\mathbb{R}$-linear, and $M_2(\mathbb{C})$ has real dimension $8$, equal to $\dim_{\mathbb{R}}\mathbb{H}_{\mathbb{D}}$, so an injective such map exists as a linear map and its image would be all of $M_2(\mathbb{C})$. What obstructs is the centre: such an embedding would give an isomorphism of $\mathbb{R}$-algebras $\mathbb{H}_{\mathbb{D}} \cong M_2(\mathbb{C})$, but the centre of $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\oplus\mathbb{H}$ is $\mathbb{R}\oplus\mathbb{R}$, not a field, while the centre of $M_2(\mathbb{C})$ is $\mathbb{C}$, a field. So the failure over $\mathbb{C}$ is a failure of the centre, not of the dimension.

Remark. Over $\mathbb{R}$ the obstruction is dimension, and over $\mathbb{C}$ it is the centre; in neither case is it the presence of zero divisors, since $M_2(\mathbb{R})$ itself has zero divisors and $\mathbb{H}_{\mathbb{D}}$ has zero divisors, so the failure is not caused by the ring-theoretic degeneracy. The smallest faithful matrix representation of $\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\oplus\mathbb{H}$ is by $4\times4$ matrices over $\mathbb{C}$ (equivalently $8\times8$ over $\mathbb{R}$), obtained by embedding each factor $\mathbb{H}$ in $M_2(\mathbb{C})$ and taking the block diagonal sum. So the dual-number algebra, although degenerate, is small enough to fit in $M_2(\mathbb{R})$, while the split biquaternion algebra, although it has the same kind of zero divisors, is too large.

Worked Examples

Example (the homomorphism property). For $A = 2 + 3\varepsilon$ and $B = 4 + 5\varepsilon$,

$$ \Phi(A)\Phi(B) = \begin{pmatrix} 2 & 3 \\ 0 & 2 \end{pmatrix}\begin{pmatrix} 4 & 5 \\ 0 & 4 \end{pmatrix} = \begin{pmatrix} 8 & 22 \\ 0 & 8 \end{pmatrix} = \Phi(AB), $$

matching $AB = 8 + 22\varepsilon$; moreover $\det\Phi(A) = 4 = N(A)$ and $\det\Phi(AB) = 64 = N(A)N(B)$.

Example (the shear subgroup). For $s \in R$ the image of $1 + s\varepsilon$ is the unipotent matrix

$$ \Phi(1 + s\varepsilon) = I + sE = \begin{pmatrix} 1 & s \\ 0 & 1 \end{pmatrix}, \qquad \Phi(1 + s\varepsilon)\Phi(1 + t\varepsilon) = \Phi\bigl(1 + (s+t)\varepsilon\bigr), $$

so these matrices form the shear subgroup $I + R\,E$, with $\Phi(1 + s\varepsilon)^{-1} = \Phi(1 - s\varepsilon)$. For instance $\Phi(1 + 3\varepsilon)\Phi(1 + 5\varepsilon) = \Phi(1 + 8\varepsilon)$.

Summary

The dual-number algebra has the faithful unital $R$-algebra representation

$$ \Phi(a + \varepsilon a') = \begin{pmatrix} a & a' \\ 0 & a \end{pmatrix}, \qquad \Phi(AB) = \Phi(A)\Phi(B), \qquad \ker\Phi = 0, $$

identifying $\mathbb{D}'_R$ with the commutative subalgebra of upper-triangular Toeplitz matrices, spanned by $I$ and the nilpotent $E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix}$ with $E^2 = 0$. The trace is $2a$, twice the real part; the determinant is $a^2 = N(A)$, the degenerate norm; and the characteristic polynomial is $(\lambda - a)^2$, exhibiting the single repeated eigenvalue equal to the real part. The distinguished submodules map to the scalar matrices $R\,I$ and the strictly upper-triangular matrices $R\,E$; dual conjugation is the inner involution by $J = \operatorname{diag}(1,-1)$; and the maximal ideal is the nilpotent line $R\,E$. The regular representation of the companion articles is the transpose convention, with the infinitesimal part in the lower-left corner. the shear subgroup is $I + R\,E$, with $\Phi(1 + s\varepsilon)\Phi(1 + t\varepsilon) = \Phi(1 + (s+t)\varepsilon)$; the algebra automorphisms $\varphi_c$ are realised by conjugation with the diagonal matrices $\operatorname{diag}(c, 1)$; and the polar form is recovered from the trace and the determinant by $B(A, B) = \tfrac{1}{2}(\operatorname{tr}\Phi(A)\operatorname{tr}\Phi(B) - \operatorname{tr}(\Phi(A)\Phi(B)))$. Comparing with the quaternion case, $\mathbb{H}$ has a faithful $2\times2$ representation over $\mathbb{C}$ with determinant the non-degenerate form $a^2 + b^2 + c^2 + d^2$, while the dual algebra has a faithful $2\times2$ representation over $R$ with determinant the degenerate form $a^2$. The split biquaternion algebra $\mathbb{H}_{\mathbb{D}}$, of real dimension eight, has no faithful $2\times2$ representation, over $\mathbb{R}$ or over $\mathbb{C}$ — by the dimension count over $\mathbb{R}$ and by the centre obstruction over $\mathbb{C}$ — the smallest faithful representation being $4\times4$ over $\mathbb{C}$.

Summary of Notation

Symbol	Meaning
$\mathbb{D}'_R = R[\varepsilon]/(\varepsilon^2)$	Dual-number algebra over $R$; $\varepsilon^2 = 0$
$\mathbb{D}' = \mathbb{D}'_{\mathbb{R}}$	Dual-number algebra over $\mathbb{R}$
$A = a + \varepsilon a'$	General dual number
$a = \operatorname{Re} A$, $a' = \operatorname{Inf} A$	Real and infinitesimal parts
$\bar{A} = a - \varepsilon a'$	Dual conjugation
$N(A) = a^2$	Norm, equal to $\det\Phi(A)$
$\Phi(a + \varepsilon a') = \begin{pmatrix} a & a' \\ 0 & a \end{pmatrix}$	Faithful matrix representation
$E = \begin{pmatrix} 0 & 1 \\ 0 & 0 \end{pmatrix} = \Phi(\varepsilon)$	Nilpotent Jordan block, $E^2 = 0$
$J = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$	Involution realizing dual conjugation
$\varphi_c$	Automorphism $\varepsilon \mapsto c\varepsilon$, conjugation by $\operatorname{diag}(c, 1)$
$B(A,B) = ab$	Polar form, $\tfrac{1}{2}(\operatorname{tr}\Phi(A)\operatorname{tr}\Phi(B) - \operatorname{tr}(\Phi(A)\Phi(B)))$
$[u] = \begin{pmatrix} a & 0 \\ b & a \end{pmatrix} = \Phi(u)^{\mathsf{T}}$	Regular representation, transpose convention
$I = \Phi(1)$	Identity matrix
$\mathbb{H}$	Quaternions, faithful $2\times2$ representation over $\mathbb{C}$
$\mathbb{H}_{\mathbb{D}} \cong \mathbb{H}\oplus\mathbb{H}$	Split biquaternions, no faithful $2\times2$ representation