Worked Examples of Exponential Polar Decomposition with Eigenbasis of Biquaternions

Introduction

This article collects the eigenbasis, the four basis elements (two idempotents and two nilpotents) and the two spinor spaces of the eight biquaternions of Worked Examples of Exponential Polar Decomposition of Biquaternions. 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 $(\tilde\Pi_+,\vec\Upsilon_-,\vec\Upsilon_+,\tilde\Pi_-)$.

In the three cases with $B=0$, the identity of case 1 and the null-vector elements of cases 6 and 8, there is no eigenbasis singled out by the element. The paragraph of the case reads the element in a fixed Peirce basis $(\tilde\Pi_{+,0},\tilde\Pi_{-,0})$ of Ideals and Peirce Decompositions of the Biquaternion Algebra; in case 1 the conclusion holds for every pair of complementary idempotents, so the fixed pair is only a choice.

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

The Eigenbasis, the Basis Matrices, and the Spinors

The basis. For $\vec Q\neq0$ put $B=\sqrt{Q_x^2+Q_y^2+Q_z^2}$, $\vec\Xi=-i\vec Q/B$ and $\tilde\Pi_\pm=\tfrac12(e_t\pm\vec\Xi)$. The two mixed corners are the complex lines $$ \mathbb{B}_{-+}=\tilde\Pi_-\mathbb{B}\tilde\Pi_+, \qquad \mathbb{B}_{+-}=\tilde\Pi_+\mathbb{B}\tilde\Pi_-, $$ and $\vec\Upsilon_+\in\mathbb{B}_{-+}$, $\vec\Upsilon_-\in\mathbb{B}_{+-}$ are their generators, normalized by $$ \vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-, \qquad \vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+, \qquad \vec\Upsilon_\pm^2=0. $$ The normalization fixes the pair up to the reciprocal rescaling $\vec\Upsilon_+\mapsto\mu\vec\Upsilon_+$, $\vec\Upsilon_-\mapsto\mu^{-1}\vec\Upsilon_-$. In the eigenbasis order $(\tilde\Pi_+,\vec\Upsilon_-,\vec\Upsilon_+,\tilde\Pi_-)$ the left-regular matrices of the four basis elements are $$ [L_{\tilde\Pi_+}]=\operatorname{diag}(1,1,0,0),\qquad [L_{\tilde\Pi_-}]=\operatorname{diag}(0,0,1,1), $$ $$ [L_{\vec\Upsilon_-}]= \begin{pmatrix}0&0&1&0\\0&0&0&1\\0&0&0&0\\0&0&0&0\end{pmatrix}, \qquad [L_{\vec\Upsilon_+}]= \begin{pmatrix}0&0&0&0\\0&0&0&0\\1&0&0&0\\0&1&0&0\end{pmatrix}, $$ and $$ [L_{\tilde Q}]=\operatorname{diag}(\lambda_+,\lambda_+,\lambda_-,\lambda_-), \qquad \lambda_\pm=Q_t\pm iB. $$ In the $2\times2$ model $\mathsf M_2^{Q}$ the same elements are $$ \mathsf M_2^{Q}(\tilde\Pi_+)=\begin{pmatrix}1&0\\0&0\end{pmatrix},\quad \mathsf M_2^{Q}(\vec\Upsilon_-)=\begin{pmatrix}0&1\\0&0\end{pmatrix},\quad \mathsf M_2^{Q}(\vec\Upsilon_+)=\begin{pmatrix}0&0\\1&0\end{pmatrix},\quad \mathsf M_2^{Q}(\tilde\Pi_-)=\begin{pmatrix}0&0\\0&1\end{pmatrix}, $$ $$ \mathsf M_2^{Q}(\tilde Q)=\begin{pmatrix}\lambda_+&0\\0&\lambda_-\end{pmatrix}. $$ When $\vec Q\parallel\vec e_z$ the eigenbasis is the standard Peirce basis of Ideals and Peirce Decompositions of the Biquaternion Algebra, §The Biquaternion Case, with the two labels exchanged according to the sign of $B$; the four matrices above are then the matrices of $\tilde\Pi_+$, $\vec\Upsilon_-$, $\vec\Upsilon_+$, $\tilde\Pi_-$ in that basis.

The spinor spaces. The two spinor spaces are the minimal left ideals $$ S_+=\mathbb{B}\tilde\Pi_+=\mathbb{C}\tilde\Pi_+\oplus\mathbb{C}\vec\Upsilon_+, \qquad S_-=\mathbb{B}\tilde\Pi_-=\mathbb{C}\vec\Upsilon_-\oplus\mathbb{C}\tilde\Pi_-. $$ Each is two-dimensional over $\mathbb{C}$. A spinor is an element of one of them, so it has two complex coordinates, not one. A spinor of first chirality is $$ \xi_+=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+, \qquad (Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})\in\mathbb{C}^2, $$ the first column of the matrix; a spinor of second chirality is $$ \xi_-=Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad (Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})\in\mathbb{C}^2, $$ the second column. The four coordinates together belong to a general element of the algebra, $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^{Q}(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}, $$ which is the pair $(\xi_+,\xi_-)$ and is not itself a spinor. The matrices of the two spinors are $$ \mathsf M_2^{Q}(\xi_+)=\begin{pmatrix}Q_{\mathrm{sp},11}&0\\Q_{\mathrm{sp},21}&0\end{pmatrix}, \qquad \mathsf M_2^{Q}(\xi_-)=\begin{pmatrix}0&Q_{\mathrm{sp},12}\\0&Q_{\mathrm{sp},22}\end{pmatrix}. $$ Under multiplication by $\tilde Q$, $$ (Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})\longmapsto(\lambda_+Q_{\mathrm{sp},11},\;\lambda_-Q_{\mathrm{sp},21}), \qquad (Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})\longmapsto(\lambda_+Q_{\mathrm{sp},12},\;\lambda_-Q_{\mathrm{sp},22}). $$

The Peirce coordinates of $\tilde Q$. For the element $\tilde Q$ of the decomposition the nilpotent coordinates vanish, $Q_{\mathrm{sp},21}=Q_{\mathrm{sp},12}=0$, and $Q_{\mathrm{sp},11}=\lambda_+$, $Q_{\mathrm{sp},22}=\lambda_-$. Its Peirce components are therefore $$ \tilde Q\tilde\Pi_+=\lambda_+\tilde\Pi_+\longleftrightarrow\begin{pmatrix}\lambda_+\\0\end{pmatrix}, \qquad \tilde Q\tilde\Pi_-=\lambda_-\tilde\Pi_-\longleftrightarrow\begin{pmatrix}0\\\lambda_-\end{pmatrix}, $$ with matrices $\operatorname{diag}(\lambda_+,0)$ and $\operatorname{diag}(0,\lambda_-)$, and their sum is $\operatorname{diag}(\lambda_+,\lambda_-)$. The zero entries are the vanishing nilpotent coordinates of this particular element, not a missing dimension: a general element in the same basis can have both coordinates nonzero in each column. For example, $\tilde Q+\vec\Upsilon_++\vec\Upsilon_-$ has the matrix $$ \mathsf M_2^{Q}(\tilde Q+\vec\Upsilon_++\vec\Upsilon_-)=\begin{pmatrix}\lambda_+&1\\1&\lambda_-\end{pmatrix}, $$ first-chirality spinor $\xi_+=( \lambda_+,1)$ and second-chirality spinor $\xi_-=(1,\lambda_-)$.

Case by case. In every case with $B\neq0$ the eigenvalues are $\lambda_\pm=Q_t\pm iB$, the Peirce components of $\tilde Q$ are $\tilde Q\tilde\Pi_+=\lambda_+\tilde\Pi_+$ and $\tilde Q\tilde\Pi_-=\lambda_-\tilde\Pi_-$; these are the columns $(\lambda_+,0)$ and $(0,\lambda_-)$ of the diagonal matrix, the nilpotent coordinates being zero because the element of the case is $\tilde Q$ itself. For cases 2 to 5 the vector part is $Q_x\vec e_x$ and $B=Q_x$, so $\vec\Xi=-i\vec e_x$ and $\tilde\Pi_\pm=\tfrac12(e_t\mp i\vec e_x)$; case 7 has a genuine three-dimensional vector part.

In case 1 the element is the identity and every pair of complementary idempotents is an eigenbasis: the two spinor spaces can be any pair of minimal ideals, and both eigenvalues are $1$. In cases 6 and 8 the vector part is null, $B=0$, so there is no eigenbasis, no normalized nilpotent pair and no spinor decomposition attached to $\tilde Q$; the multiplication is a single Jordan block with the repeated eigenvalue $1$, respectively $1+i$, and the matrix is not diagonalisable.

The Eight Cases

Case 1: $e_t$

The four basis elements. $B=0$, so there is no eigenbasis. The fixed Peirce basis of the algebra is $$ \tilde\Pi_+=\tfrac12(e_t+i\vec e_z),\quad \tilde\Pi_-=\tfrac12(e_t-i\vec e_z),\quad \vec\Upsilon_+=\tfrac12(i\vec e_x+\vec e_y),\quad \vec\Upsilon_-=\tfrac12(i\vec e_x-\vec e_y), $$ with $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. $\tilde Q=e_t=\tilde\Pi_++\tilde\Pi_-$, so $\lambda_+=\lambda_-=1$ and $\mathsf M_2^Q(\tilde Q)=\mathrm I_2$; every complementary pair of idempotents gives the same decomposition.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the coordinates are $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},12},Q_{\mathrm{sp},21},Q_{\mathrm{sp},22})=(1,0,0,1)$.

Case 2: $\vec e_x$

The four basis elements. $\vec Q=\vec e_x$ and $B=1$, so $\vec\Xi=-i\vec e_x$ and $$ \tilde\Pi_+=\tfrac12(e_t-i\vec e_x),\quad \tilde\Pi_-=\tfrac12(e_t+i\vec e_x),\quad \vec\Upsilon_+=\tfrac12(\vec e_y+i\vec e_z),\quad \vec\Upsilon_-=\tfrac12(i\vec e_z-\vec e_y), $$ with $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. $\tilde Q=i\tilde\Pi_+-i\tilde\Pi_-$, so $\lambda_+=i$, $\lambda_-=-i$ and $\mathsf M_2^Q(\tilde Q)=\operatorname{diag}(i,-i)$.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the coordinates are $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},12},Q_{\mathrm{sp},21},Q_{\mathrm{sp},22})=(i,0,0,-i)$.

Case 3: $\tfrac{1}{\sqrt2}(e_t+\vec e_x)$

The four basis elements. The vector part is again along $\vec e_x$, so $\vec\Xi=-i\vec e_x$ and the four basis elements are those of Case 2: $$ \tilde\Pi_+=\tfrac12(e_t-i\vec e_x),\quad \tilde\Pi_-=\tfrac12(e_t+i\vec e_x),\quad \vec\Upsilon_+=\tfrac12(\vec e_y+i\vec e_z),\quad \vec\Upsilon_-=\tfrac12(i\vec e_z-\vec e_y), $$ with $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. $\tilde Q=\tfrac{1}{\sqrt2}(e_t+\vec e_x)=\lambda_+\tilde\Pi_++\lambda_-\tilde\Pi_-$, with $\lambda_+=\tfrac{1+i}{\sqrt2}$ and $\lambda_-=\tfrac{1-i}{\sqrt2}$, so $\mathsf M_2^Q(\tilde Q)=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the coordinates are $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},12},Q_{\mathrm{sp},21},Q_{\mathrm{sp},22})=(\lambda_+,0,0,\lambda_-)$.

Case 4: $\sqrt2\,e_t+i\vec e_x$

The four basis elements. The vector part is again along $\vec e_x$, so $\vec\Xi=-i\vec e_x$ and the four basis elements are those of Case 2: $$ \tilde\Pi_+=\tfrac12(e_t-i\vec e_x),\quad \tilde\Pi_-=\tfrac12(e_t+i\vec e_x),\quad \vec\Upsilon_+=\tfrac12(\vec e_y+i\vec e_z),\quad \vec\Upsilon_-=\tfrac12(i\vec e_z-\vec e_y), $$ with $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. $\tilde Q=\sqrt2\,e_t+i\vec e_x=\lambda_+\tilde\Pi_++\lambda_-\tilde\Pi_-$, with $\lambda_+=\sqrt2-1$ and $\lambda_-=\sqrt2+1$, so $\mathsf M_2^Q(\tilde Q)=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the coordinates are $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},12},Q_{\mathrm{sp},21},Q_{\mathrm{sp},22})=(\lambda_+,0,0,\lambda_-)$.

Case 5: $\bigl(1-\tfrac{i}{\sqrt2}\bigr)e_t+\bigl(1+\tfrac{i}{\sqrt2}\bigr)\vec e_x$

The four basis elements. The vector part is again along $\vec e_x$, so $\vec\Xi=-i\vec e_x$ and the four basis elements are those of Case 2: $$ \tilde\Pi_+=\tfrac12(e_t-i\vec e_x),\quad \tilde\Pi_-=\tfrac12(e_t+i\vec e_x),\quad \vec\Upsilon_+=\tfrac12(\vec e_y+i\vec e_z),\quad \vec\Upsilon_-=\tfrac12(i\vec e_z-\vec e_y), $$ with $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. $\tilde Q=\bigl(1-\tfrac{i}{\sqrt2}\bigr)e_t+\bigl(1+\tfrac{i}{\sqrt2}\bigr)\vec e_x=\lambda_+\tilde\Pi_++\lambda_-\tilde\Pi_-$, with $\lambda_+=\bigl(1-\tfrac{1}{\sqrt2}\bigr)(1+i)$ and $\lambda_-=\bigl(1+\tfrac{1}{\sqrt2}\bigr)(1-i)$, so $\mathsf M_2^Q(\tilde Q)=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the coordinates are $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},12},Q_{\mathrm{sp},21},Q_{\mathrm{sp},22})=(\lambda_+,0,0,\lambda_-)$.

Case 6: $e_t+\vec e_y+\tfrac{i}{\sqrt2}\vec e_x+\tfrac{i}{\sqrt2}\vec e_z$

The four basis elements. $B=0$, so there is no eigenbasis. The fixed Peirce basis of the algebra is $$ \tilde\Pi_+=\tfrac12(e_t+i\vec e_z),\quad \tilde\Pi_-=\tfrac12(e_t-i\vec e_z),\quad \vec\Upsilon_+=\tfrac12(i\vec e_x+\vec e_y),\quad \vec\Upsilon_-=\tfrac12(i\vec e_x-\vec e_y), $$ with $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. The eigenvalue is $\lambda_+=1$, double, and $$ \mathsf M_2^Q(\tilde Q)=\begin{pmatrix}1+\tfrac{1}{\sqrt2}&\tfrac{1}{\sqrt2}-1\\[2pt]1+\tfrac{1}{\sqrt2}&1-\tfrac{1}{\sqrt2}\end{pmatrix}=\mathrm I_2+\begin{pmatrix}\tfrac{1}{\sqrt2}&\tfrac{1}{\sqrt2}-1\\[2pt]1+\tfrac{1}{\sqrt2}&-\tfrac{1}{\sqrt2}\end{pmatrix}, $$ and the second matrix is square-zero. The element is unipotent: it is not diagonalisable and no eigenbasis exists.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the two columns are $(1+\tfrac{1}{\sqrt2},1+\tfrac{1}{\sqrt2})$ and $(\tfrac{1}{\sqrt2}-1,1-\tfrac{1}{\sqrt2})$; left multiplication by $\tilde Q$ acts on them by $\mathsf M_2^Q(\tilde Q)$, not diagonally.

Case 7: $\bigl(1-\tfrac{i}{2}\bigr)e_t+\tfrac{1+i}{\sqrt2}\vec e_x+\tfrac{1}{\sqrt2}\vec e_y+\tfrac{i}{2}\vec e_z$

The four basis elements. $B=\sqrt{i+\tfrac14}$ and $\vec\Xi=-i\vec Q/B$, so $$ \tilde\Pi_\pm=\frac{e_t\pm\vec\Xi}{2},\qquad \vec W=\frac{\vec Q\times\vec e_x}{|\vec Q\times\vec e_x|}=i\vec e_y-\sqrt2\,\vec e_z,\qquad \vec\Upsilon_\pm=-i\,\vec W\tilde\Pi_\pm, $$ where $\vec W\cdot\vec Q=0$ and $\vec W\cdot\vec W=1$; then $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. $\tilde Q=\bigl(1-\tfrac{i}{2}\bigr)e_t+\tfrac{1+i}{\sqrt2}\vec e_x+\tfrac{1}{\sqrt2}\vec e_y+\tfrac{i}{2}\vec e_z=\lambda_+\tilde\Pi_++\lambda_-\tilde\Pi_-$ with $\lambda_\pm=1-\tfrac{i}{2}\pm iB$, so $\mathsf M_2^Q(\tilde Q)=\operatorname{diag}(\lambda_+,\lambda_-)$.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the coordinates are $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},12},Q_{\mathrm{sp},21},Q_{\mathrm{sp},22})=(\lambda_+,0,0,\lambda_-)$.

Case 8: $(1+i)e_t+\vec e_x+i\vec e_y$

The four basis elements. $B=0$, so there is no eigenbasis. The fixed Peirce basis of the algebra is $$ \tilde\Pi_+=\tfrac12(e_t+i\vec e_z),\quad \tilde\Pi_-=\tfrac12(e_t-i\vec e_z),\quad \vec\Upsilon_+=\tfrac12(i\vec e_x+\vec e_y),\quad \vec\Upsilon_-=\tfrac12(i\vec e_x-\vec e_y), $$ with $\tilde\Pi_\pm^2=\tilde\Pi_\pm$, $\vec\Upsilon_\pm^2=0$, $\vec\Upsilon_+\vec\Upsilon_-=\tilde\Pi_-$ and $\vec\Upsilon_-\vec\Upsilon_+=\tilde\Pi_+$.

The eigenvalue decomposition. The eigenvalue is $\lambda_+=1+i$, double, and $$ \mathsf M_2^Q(\tilde Q)=\begin{pmatrix}1+i&-2i\\0&1+i\end{pmatrix}=(1+i)\mathrm I_2+\begin{pmatrix}0&-2i\\0&0\end{pmatrix}, $$ and the second matrix is square-zero. The element is not diagonalisable and no eigenbasis exists.

The spinor spaces and a general element. $S_+=\mathbb B\tilde\Pi_+=\mathbb C\tilde\Pi_+\oplus\mathbb C\vec\Upsilon_+$ and $S_-=\mathbb B\tilde\Pi_-=\mathbb C\vec\Upsilon_-\oplus\mathbb C\tilde\Pi_-$, and a general element of $\mathbb B$ is $$ \tilde Q_{\mathrm{sp}}=Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},21}\vec\Upsilon_++Q_{\mathrm{sp},22}\tilde\Pi_-, \qquad \mathsf M_2^Q(\tilde Q_{\mathrm{sp}})=\begin{pmatrix}Q_{\mathrm{sp},11}&Q_{\mathrm{sp},12}\\Q_{\mathrm{sp},21}&Q_{\mathrm{sp},22}\end{pmatrix}. $$ The first column $(Q_{\mathrm{sp},11},Q_{\mathrm{sp},21})$ is a spinor of first chirality, $Q_{\mathrm{sp},11}\tilde\Pi_++Q_{\mathrm{sp},21}\vec\Upsilon_+$; the second column $(Q_{\mathrm{sp},12},Q_{\mathrm{sp},22})$ is a spinor of second chirality, $Q_{\mathrm{sp},12}\vec\Upsilon_-+Q_{\mathrm{sp},22}\tilde\Pi_-$. For $\tilde Q$ itself the two columns are $(1+i,0)$ and $(-2i,1+i)$; left multiplication by $\tilde Q$ acts on them by $\mathsf M_2^Q(\tilde Q)$, not diagonally.

Summary

• The eigenbasis depends on the element through its root $\vec\Xi$; the two idempotents change from case to case, unlike the fixed idempotents of the split-complex and reduced-biquaternion layers.
• In every case with $B\neq0$ the eigenvalues are $\lambda_\pm=Q_t\pm iB$, the Peirce components of $\tilde Q$ are $\lambda_+\tilde\Pi_+$ and $\lambda_-\tilde\Pi_-$; these are the two columns of the diagonal matrix.
• In case 1 the element is the identity, every pair of complementary idempotents is an eigenbasis, and the two spinor spaces can be any pair of minimal ideals.
• In cases 6 and 8 the vector part is null, $B=0$: there is no eigenbasis, no normalized nilpotent pair and no spinor decomposition attached to $\tilde Q$, and the multiplication is a single Jordan block.

Summary of Notation

Further Reading

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