Worked Examples of Exponential Polar Decomposition with Eigenbasis of Reduced Biquaternions

Introduction

This article collects the eigenbasis, the idempotents and the two spinor spaces of the eight reduced biquaternions of Worked Examples of Exponential Polar Decomposition of Reduced Biquaternions. The polar decomposition itself, with its two components, 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

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

In the standard basis $(1,j)$ the first four are $[L_1]=\mathrm I_2$, $[L_j]=\sigma_x=\begin{pmatrix}0&1\\1&0\end{pmatrix}$, $[L_i]=i\mathrm I_2$ and $[L_{ij}]=i\sigma_x$; the matrix picture $\mathsf M(\breve Q)$ is the matrix in that basis, and the diagonal form below is its conjugate. For $\breve Q=Q_{\blacktriangle}+jQ_{\blacktriangledown}$,

$$ [L_{\breve Q}]=\begin{pmatrix}C_+&0\\0&C_-\end{pmatrix}, \qquad C_\pm=Q_{\blacktriangle}\pm Q_{\blacktriangledown}, $$

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

The spinor spaces. The two spinor spaces are the minimal ideals $S_\pm=\mathbb{C}\Pi_\pm$, each a complex line; the algebra is commutative, so there are no nilpotents and no mixed part. A general element is $\breve Q_{\mathrm{sp}}=A_1\Pi_++A_2\Pi_-$, with $A_1,A_2\in\mathbb{C}$ and $$ [L_{\breve Q_{\mathrm{sp}}}]=\begin{pmatrix}A_1&0\\0&A_2\end{pmatrix}. $$ 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 $\breve Q$ is the scalar $C_+$ and on $S_-$ it is the scalar $C_-$.

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 $\breve Q=C_+\Pi_++C_-\Pi_-$, so the Peirce components are $\breve Q\Pi_+=C_+\Pi_+$ and $\breve Q\Pi_-=C_-\Pi_-$, and the matrix is diagonal.

In cases 2, 3 and 7 the two eigenvalues coincide, so the element is a complex scalar and the matrix is a multiple of the identity; 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: $3+4i+j(1-i)$

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. $\breve Q=3+4i+j(1-i)=C_+\Pi_++C_-\Pi_-$, with $C_+=4+3i$ and $C_-=2+5i$, so $[L_{\breve Q}]=\operatorname{diag}(C_+,C_-)$.

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

Case 2: $1+i$

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. $\breve Q=1+i=C_+\Pi_++C_-\Pi_-$, with $C_+=1+i$ and $C_-=1+i$, so $[L_{\breve Q}]=\operatorname{diag}(C_+,C_-)$.

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

Case 3: $i$

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. $\breve Q=i=C_+\Pi_++C_-\Pi_-$, with $C_+=i$ and $C_-=i$, so $[L_{\breve Q}]=\operatorname{diag}(C_+,C_-)$.

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

Case 4: $ij$

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. $\breve Q=ij=C_+\Pi_++C_-\Pi_-$, with $C_+=i$ and $C_-=-i$, so $[L_{\breve Q}]=\operatorname{diag}(C_+,C_-)$.

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

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. $\breve Q=j=C_+\Pi_++C_-\Pi_-$, with $C_+=1$ and $C_-=-1$, so $[L_{\breve Q}]=\operatorname{diag}(C_+,C_-)$.

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

Case 6: $\tfrac54+\tfrac34j$

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. $\breve Q=\tfrac54+\tfrac34j=C_+\Pi_++C_-\Pi_-$, with $C_+=2$ and $C_-=\tfrac12$, so $[L_{\breve Q}]=\operatorname{diag}(C_+,C_-)$.

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

Case 7: $-1$

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. $\breve Q=-1=C_+\Pi_++C_-\Pi_-$, with $C_+=-1$ and $C_-=-1$, so $[L_{\breve Q}]=\operatorname{diag}(C_+,C_-)$.

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

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

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

Summary

• The eigenbasis $(\Pi_+,\Pi_-)$ is the same in all eight cases, because the idempotents do not depend on the element; only the eigenvalues $C_\pm$ change.
• In cases 2, 3 and 7 the two eigenvalues coincide, so the element is a complex scalar, 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 $\breve Q\Pi_-$ vanishes and the matrix has rank one.
• The two Peirce components sum to $[L_{\breve Q}]$ in every case: $\breve Q\Pi_++\breve Q\Pi_-=C_+\Pi_++C_-\Pi_-$ is the diagonal matrix $\operatorname{diag}(C_+,C_-)$.

Summary of Notation

Further Reading

• Worked Examples of Exponential Polar Decomposition of Reduced Biquaternions, for the polar decomposition, its word and its check.
• Relations between Exponential Polar Decomposition and Eigenbasis a Reduced Biquaternion, for the relation between the two decompositions in the commutative four-dimensional case.