Worked Examples of Exponential Polar Decomposition with Eigenbasis of Split-Complex Numbers

Introduction

This article collects the eigenbasis, the idempotents and the two spinor spaces of the eight split-complex numbers of Worked Examples of Exponential Polar Decomposition of Split-Complex Numbers. The polar decomposition itself, with its coordinates, its word and its check, is in that article; here the same eight elements are read in the eigenbasis $(\Pi_+,\Pi_-)$, with $\Pi_\pm=\tfrac12(1\pm j)$.

The eight elements, in the order of the cases, are collected in the table.

The Eigenbasis, the Basis Matrices, and the Spinors

The basis matrices. In the eigenbasis $(\Pi_+,\Pi_-)$, with $\Pi_\pm=\tfrac12(1\pm j)$, multiplication by each basis element has the matrix

$$ [L_1]=\begin{pmatrix}1&0\\0&1\end{pmatrix},\qquad [L_j]=\begin{pmatrix}1&0\\0&-1\end{pmatrix},\qquad [L_{\Pi_+}]=\begin{pmatrix}1&0\\0&0\end{pmatrix},\qquad [L_{\Pi_-}]=\begin{pmatrix}0&0\\0&1\end{pmatrix}. $$

In the standard basis $(1,j)$ the first two are $[L_1]=\mathrm I_2$ and $[L_j]=\begin{pmatrix}0&1\\1&0\end{pmatrix}$; the two pairs are conjugate by the change of basis whose columns are the coordinates of $\Pi_+$ and $\Pi_-$. For $Q=q+jq'$,

$$ [L_Q]=\begin{pmatrix}\lambda_+&0\\0&\lambda_-\end{pmatrix}, \qquad \lambda_\pm=q\pm q', $$

which is the matrix form of $Q=\lambda_+\Pi_++\lambda_-\Pi_-$.

The spinor spaces. The two spinor spaces are the minimal ideals $S_\pm=\mathbb{R}\Pi_\pm$, each a real line; the algebra is commutative, so there are no nilpotents and no mixed part. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $a_1,a_2\in\mathbb{R}$ and $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}. $$ Its first column $a_1\Pi_+$ is a spinor of first chirality and its second column $a_2\Pi_-$ a spinor of second chirality. On $S_+$ multiplication by $Q$ is the scalar $\lambda_+$ and on $S_-$ it is the scalar $\lambda_-$. After complexification the same formulas hold with $a_1,a_2\in\mathbb{C}$.

Case by case. The two basis matrices $[L_{\Pi_+}]$ and $[L_{\Pi_-}]$ are the same in all eight cases, because the idempotents do not depend on the element. In every case $Q=\lambda_+\Pi_++\lambda_-\Pi_-$, so the Peirce components are $Q\Pi_+=\lambda_+\Pi_+$ and $Q\Pi_-=\lambda_-\Pi_-$, and the matrix is diagonal.

In case 6 and case 7 the two eigenvalues coincide and the two lines carry the same scale; in case 8 the second eigenvalue is zero and the matrix has rank one.

The Eight Cases

Case 1: $5+3j$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=5+3j=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=8$ and $\lambda_-=2$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Case 2: $3+5j$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=3+5j=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=8$ and $\lambda_-=-2$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Case 3: $-5-3j$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=-5-3j=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=-8$ and $\lambda_-=-2$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Case 4: $1-2j$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=1-2j=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=-1$ and $\lambda_-=3$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Case 5: $j$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=j=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=1$ and $\lambda_-=-1$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Case 6: $2$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=2=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=2$ and $\lambda_-=2$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Case 7: $-2$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=-2=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=-2$ and $\lambda_-=-2$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Case 8: $1+j$

The two idempotents. $\Pi_\pm=\tfrac12(1\pm j)$. The algebra is commutative, so the mixed Peirce corners vanish: there are no nilpotents, and the Peirce basis has two elements, not four.

The eigenvalue decomposition. $Q=1+j=\lambda_+\Pi_++\lambda_-\Pi_-$, with $\lambda_+=2$ and $\lambda_-=0$, so $[L_{Q}]=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_\pm=\mathbb R\Pi_\pm$, one line each, and the algebra is $S_+\oplus S_-$. A general element is $Q_{\mathrm{sp}}=a_1\Pi_++a_2\Pi_-$, with $$ [L_{Q_{\mathrm{sp}}}]=\begin{pmatrix}a_1&0\\0&a_2\end{pmatrix}; $$ the first column $a_1\Pi_+$ is a spinor of first chirality and the second column $a_2\Pi_-$ a spinor of second chirality. For $Q$ itself the coordinates are $(a_1,a_2)=(\lambda_+,\lambda_-)$.

Summary

• The eigenbasis $(\Pi_+,\Pi_-)$ is the same in all eight cases, because the idempotents do not depend on the element; only the eigenvalues $\lambda_\pm$ change.
• In cases 6 and 7 the two eigenvalues coincide, so the two spinor lines carry the same scale and the matrix is a multiple of the identity.
• In case 8 the second eigenvalue is zero; the component $Q\Pi_-$ vanishes and the matrix has rank one.
• The two Peirce components sum to $[L_Q]$ in every case: $Q\Pi_++Q\Pi_-=\lambda_+\Pi_++\lambda_-\Pi_-$ is the diagonal matrix $\operatorname{diag}(\lambda_+,\lambda_-)$.

Summary of Notation

Further Reading

• Worked Examples of Exponential Polar Decomposition of Split-Complex Numbers, for the polar decomposition, its word and its check.
• Relations between Exponential Polar Decomposition and Eigenbasis of a Split-Complex Number, for the relation between the two decompositions in the real two-dimensional case.