The Polar Decomposition of an N-Degenerate Biquaternion

Introduction

The general polar decomposition of The Polar Decomposition of a Biquaternion writes every biquaternion as

$$ \tilde Q=\tilde Q_H\,\tilde Q_0, $$

with $\tilde Q_H$ Hermitian positive semidefinite and $\tilde Q_0$ a partial isometry. The companion article The Polar Decomposition of an N-Regular Biquaternion treated the case $\langle\tilde Q\rangle_{\natural}\neq0$, where the factors sharpen: $\tilde Q_H$ is positive definite, $\tilde Q_0$ is unitary, and the pair refines into the four-factor word $\tilde Q=r\zeta\tilde U_{\mathrm b}\tilde u_{\mathrm r}$.

This article treats the complementary case $\langle\tilde Q\rangle_{\natural}=0$. The modulus vanishes, so the modulus–unit form $\tilde Q=\Rho\tilde U$ with $\Rho\neq0$ does not exist, and neither does the four-factor word, whose product has norm $\Rho^{2}=0$ and cannot represent a nonzero element. What survives is the two-factor decomposition: $\tilde Q_H$ is positive semidefinite of rank at most one, and $\tilde Q_0$ is a partial isometry that is not unitary.

The article states why the modulus and the four factors fail (§The Vanishing of the Modulus), records the surviving pair (§The Two-Factor Decomposition), computes the two factors in closed form (§The Hermitian Factor, §The Partial-Isometry Factor), describes the two sectors of the null cone (§The Two Sectors), and gives four worked examples. The general pair, its uniqueness and the continuity of its Hermitian factor are established in The Polar Decomposition of a Biquaternion and are cited, not proved again.

The conventions and the symbols are those of Conventions in Mathematics, N-Regularity and Degeneracy in the Biquaternion Algebra and The Four Forms and the Four Algebraic Norms over the Biquaternion Algebra; the symbols are collected in §Summary of Notation. The product is the plain product and the conjugation is ${}^{*}=\bar{\cdot}\circ{}^{\natural}$.

The Vanishing of the Modulus

Let $\tilde Q$ be N-degenerate, $\langle\tilde Q\rangle_{\natural}=0$. The modulus of the N-regular article is a square root of the norm,

$$ \Rho^{2}=\langle\tilde Q\rangle_{\natural}=0, $$

so $\Rho=0$. The modulus–unit decomposition requires $\Rho\neq0$, since it divides by the modulus; at $\Rho=0$ the unit factor $\tilde Q/\Rho$ is undefined.

The four-factor word fails for the same reason, and the failure is visible in the norm. Its factors are constrained by

$$ r=\lvert\Rho\rvert>0,\qquad \lvert\zeta\rvert=1,\qquad \langle\tilde U_{\mathrm b}\rangle_{\natural}=1,\qquad \langle\tilde u_{\mathrm r}\rangle_{\natural}=1, $$

and the norm is multiplicative, so

$$ \bigl\langle r\zeta\tilde U_{\mathrm b}\tilde u_{\mathrm r}\bigr\rangle_{\natural}=r^{2}\zeta^{2}\langle\tilde U_{\mathrm b}\rangle_{\natural}\langle\tilde u_{\mathrm r}\rangle_{\natural}=\Rho^{2}=0 . $$

A product of norm zero cannot equal a nonzero element. At the origin the failure is stronger: $\Rho=0$ forces $r=0$, which contradicts $r>0$.

The modulus–unit form and the four-factor word therefore exist exactly on the N-regular elements. The N-degenerate elements are the null cone

$$ \mathcal N=\{\tilde Q:\langle\tilde Q\rangle_{\natural}=0\}, $$

one complex equation, hence two real equations, of real codimension two (N-Regularity and Degeneracy in the Biquaternion Algebra, §The Zero Divisors).

The Two-Factor Decomposition

The general pair survives on the null cone. Every N-degenerate element is

$$ \boxed{\;\tilde Q=\tilde Q_H\,\tilde Q_0,\qquad \tilde Q_H=\bigl(\tilde Q\tilde Q^{*}\bigr)^{1/2},\quad \tilde Q_0=\tilde Q_H^{+}\tilde Q,\;} $$

with ${}^{+}$ the Moore–Penrose inverse. The pair exists for every element of the algebra and is unique (The Polar Decomposition of a Biquaternion, §Uniqueness); on the null cone it takes the following form.

Proposition. If $\langle\tilde Q\rangle_{\natural}=0$, then $\tilde Q_H$ is Hermitian positive semidefinite of rank at most one, $\tilde Q_0$ is a partial isometry, and $\tilde Q_0$ is not unitary for $\tilde Q\neq0$. At the origin both factors vanish.

The rank is one exactly when $\tilde Q\neq0$, and zero at the origin. The partial isometry is not unitary because its final projection $\tilde Q_0\tilde Q_0^{*}=\tilde Q_H/(2a)$ is a rank-one projection, not the identity, for $a>0$. The two factors are functions of $\tilde Q$ alone. The Hermitian factor is the continuous limit of the N-regular Hermitian factor as the element approaches the cone; the isometry factor is not continuous there (The Polar Decomposition of a Biquaternion, §Continuity).

The Hermitian Factor

Write the element and its Hermitian square in the coordinates of The Polar Decomposition of a Biquaternion, §Notation:

$$ \tilde Q=Q_te_t+Q_x\vec e_x+Q_y\vec e_y+Q_z\vec e_z,\qquad \tilde Q\tilde Q^{*}=\langle\tilde Q\rangle_{*}e_t+i\vec W, $$

with $\vec W=W_x\vec e_x+W_y\vec e_y+W_z\vec e_z$ real and

$$ W_x=2\bigl(\operatorname{Im}(Q_x\overline{Q_t})-\operatorname{Im}(Q_y\overline{Q_z})\bigr),\quad W_y=2\bigl(\operatorname{Im}(Q_y\overline{Q_t})-\operatorname{Im}(Q_z\overline{Q_x})\bigr),\quad W_z=2\bigl(\operatorname{Im}(Q_z\overline{Q_t})-\operatorname{Im}(Q_x\overline{Q_y})\bigr). $$

The two identities of that section become, at $\langle\tilde Q\rangle_{\natural}=0$,

$$ \bigl\langle\tilde Q\tilde Q^{*}\bigr\rangle_{\natural}=\langle\tilde Q\rangle_{*}^{2}-\lvert\vec W\rvert^{2}=\bigl\lvert\langle\tilde Q\rangle_{\natural}\bigr\rvert^{2}=0, $$

so

$$ \lvert\vec W\rvert=\langle\tilde Q\rangle_{*},\qquad a=\sqrt{\frac{\langle\tilde Q\rangle_{*}}{2}},\qquad \lvert\vec W\rvert=2a^{2}. $$

The number $a$ is real and non-negative, and $a=0$ exactly when $\tilde Q=0$.

Theorem. For $\tilde Q\neq0$ on the null cone,

$$ \boxed{\;\tilde Q_H=a\,e_t+\frac{i}{2a}\,\vec W\;},\qquad a=\sqrt{\frac{\langle\tilde Q\rangle_{*}}{2}}, $$

and $\tilde Q_H=0$ at the origin.

Proof. The element is Hermitian, since $a$ is real and $i\vec W$ is Hermitian. Squaring and using $\vec W^{2}=-\lvert\vec W\rvert^{2}e_t$,

$$ \tilde Q_H^{2}=\Bigl(a^{2}+\frac{\lvert\vec W\rvert^{2}}{4a^{2}}\Bigr)e_t+i\vec W =2a^{2}e_t+i\vec W =\langle\tilde Q\rangle_{*}e_t+i\vec W =\tilde Q\tilde Q^{*}, $$

the third equality by $\lvert\vec W\rvert=2a^{2}$. The spectral values of $\tilde Q_H$ are $a\pm\lvert\vec W\rvert/(2a)=a\pm a$, that is, $0$ and $2a$, so the element is positive semidefinite of rank one. It is the unique square root of $\tilde Q\tilde Q^{*}$ in the cone, by the uniqueness of that root. $\square$

The scalar part of $\tilde Q_H$ is $a$, and its vector part is purely imaginary of length $\lvert\vec W\rvert/(2a)=a$. The factor is the sum of the central element $a e_t$ and a vector part of the same length; its two spectral values are $0$ and $2a$.

The Partial-Isometry Factor

The Moore–Penrose inverse of the Hermitian factor follows from the quadratic identity it satisfies. Since $\tilde Q_H^{2}=2a\tilde Q_H$ and $a>0$,

$$ \tilde Q_H^{3}=4a^{2}\tilde Q_H, \qquad \tilde Q_H^{+}=\frac{\tilde Q_H}{4a^{2}} . $$

The four Penrose conditions are checked by direct multiplication, as in The Polar Decomposition of a Biquaternion, §The Moore–Penrose Inverse of the Hermitian Factor. At the origin the inverse is $0$.

Theorem. For $\tilde Q\neq0$ on the null cone the isometry factor is

$$ \boxed{\;\tilde Q_0=\tilde Q_H^{+}\tilde Q=\frac{\tilde Q_H\,\tilde Q}{4a^{2}}\;} $$

and it is a partial isometry of rank one:

$$ \tilde Q_0\tilde Q_0^{*}\tilde Q_0=\tilde Q_0,\qquad \tilde Q_0^{*}\tilde Q_0\tilde Q_0^{*}=\tilde Q_0^{*},\qquad \tilde Q_0\tilde Q_0^{*}=\frac{\tilde Q_H}{2a}. $$

Proof. The product $\tilde Q_0\tilde Q_0^{*}$ is

$$ \tilde Q_H^{+}\tilde Q\tilde Q^{*}\tilde Q_H^{+}=\tilde Q_H^{+}\tilde Q_H^{2}\tilde Q_H^{+}=\tilde Q_H\tilde Q_H^{+}=\frac{\tilde Q_H}{2a}, $$

using $\tilde Q_H\tilde Q_H^{+}=\tilde Q_H/(2a)$, which is the final projection of the Hermitian factor. This product is a rank-one projection, so the first Penrose identity gives $\tilde Q_0\tilde Q_0^{*}\tilde Q_0=\tilde Q_0$, and the second follows symmetrically. $\square$

The factor $\tilde Q_0$ is not unitary: its product with its conjugate is the rank-one projection $\tilde Q_H/(2a)$, not $e_t$. At the origin, $\tilde Q_H=0$, $\tilde Q_H^{+}=0$ and $\tilde Q_0=0$.

The Two Sectors

The null cone splits into the two sectors of the spectral problem, $\Sigma_{01}$ and $\Sigma_{00}$.

The sector $\Sigma_{01}$

$$ \Sigma_{01}=\{\langle\tilde Q\rangle_{\natural}=0,\ B^{2}\neq0\},\qquad B^{2}=Q_x^{2}+Q_y^{2}+Q_z^{2}. $$

It is The N-Degenerate, Spectrally Regular Sector. The characteristic polynomial is $X^{2}-2Q_tX$, with the two distinct roots $0$ and $2Q_t\neq0$, and the element has the spectral decomposition $\tilde Q=c\,\tilde\Pi$ with $c\neq0$ and $\tilde\Pi$ an idempotent different from $0$ and $e_t$. The idempotent need not be Hermitian. The Hermitian factor is positive semidefinite of rank one, the isometry factor is a partial isometry of rank one, and both are computed by the formulas above.

The sector $\Sigma_{00}$

$$ \Sigma_{00}=\{\langle\tilde Q\rangle_{\natural}=0,\ B^{2}=0\}. $$

It is The N-Degenerate, Spectrally Degenerate Sector. The two conditions give $Q_t=0$ and $Q_x^{2}+Q_y^{2}+Q_z^{2}=0$, so the element is pure and square-zero:

$$ \tilde Q=\vec Q=Q_x\vec e_x+Q_y\vec e_y+Q_z\vec e_z,\qquad \tilde Q^{2}=0 . $$

The sector splits into the origin, where $\tilde Q_H=0$ and $\tilde Q_0=0$, and the nonzero nilpotents, where $\tilde Q_H$ has rank one and $\tilde Q_0$ is a partial isometry of rank one. The nonzero nilpotents are the pure elements of the nilpotent cone (Square Roots of Zero and the Nilpotents of the Biquaternion Algebra).

The two sectors are disjoint. The origin lies in $\Sigma_{00}$, and it is the only element at which $a=0$. On both sectors the two-factor decomposition exists, and on neither does the four-factor word exist.

Examples

Example 1: the sector $\Sigma_{01}$

$$ \tilde Q=e_t+i\vec e_x . $$

The norm is $\langle\tilde Q\rangle_{\natural}=1+i^{2}=0$, the Hermitian norm is $\langle\tilde Q\rangle_{*}=2$, so $a=1$, and $\vec W=2\vec e_x$. Then

$$ \tilde Q_H=e_t+i\vec e_x,\qquad \tilde Q_H^{+}=\frac{e_t+i\vec e_x}{4},\qquad \tilde Q_0=\frac{e_t+i\vec e_x}{2}. $$

The decomposition reads

$$ e_t+i\vec e_x=\bigl(e_t+i\vec e_x\bigr)\cdot\frac{e_t+i\vec e_x}{2}. $$

The factor $\tilde Q_0$ is a projection, hence a partial isometry, but not unitary: $\tilde Q_0\tilde Q_0^{*}=\tilde Q_0$ is a rank-one projection, not the identity.

Example 2: the sector $\Sigma_{01}$

$$ \tilde Q=e_t+\vec e_x+i\vec e_y+i\vec e_z . $$

The norm is $\langle\tilde Q\rangle_{\natural}=1+1+i^{2}+i^{2}=0$, the Hermitian norm is $4$, so $a=\sqrt2$, and $\vec W=4\vec e_z$. Then

$$ \tilde Q_H=\sqrt2\,e_t+i\sqrt2\,\vec e_z,\qquad \tilde Q_0=\frac{\tilde Q}{2\sqrt2}. $$

The decomposition reads

$$ e_t+\vec e_x+i\vec e_y+i\vec e_z =\bigl(\sqrt2\,e_t+i\sqrt2\,\vec e_z\bigr)\cdot\frac{\tilde Q}{2\sqrt2}. $$

The factor $\tilde Q_0$ is a partial isometry of rank one, not unitary.

Example 3: the sector $\Sigma_{00}$, a nonzero nilpotent

$$ \tilde Q=\vec e_x+i\vec e_y . $$

The norm is $\langle\tilde Q\rangle_{\natural}=1+i^{2}=0$, and $B^{2}=1+i^{2}=0$, so $\tilde Q^{2}=0$. The Hermitian norm is $2$, so $a=1$, and $\vec W=2\vec e_z$. Then

$$ \tilde Q_H=e_t+i\vec e_z,\qquad \tilde Q_H^{+}=\frac{e_t+i\vec e_z}{4},\qquad \tilde Q_0=\frac{\vec e_x+i\vec e_y}{2}. $$

The decomposition reads

$$ \vec e_x+i\vec e_y=\bigl(e_t+i\vec e_z\bigr)\cdot\frac{\vec e_x+i\vec e_y}{2}. $$

The Hermitian factor has rank one, and the isometry factor is a partial isometry of rank one, not unitary.

Example 4: the origin

$$ \tilde Q=0 . $$

Then $\langle\tilde Q\rangle_{\natural}=0$, $B^{2}=0$, and

$$ \tilde Q_H=0,\qquad \tilde Q_H^{+}=0,\qquad \tilde Q_0=0 . $$

The decomposition reads $0=0\cdot0$.

Summary

On the null cone the modulus–unit form and the four-factor word do not exist: the modulus vanishes, and the norm of the four-factor product is $\Rho^{2}=0$, so it cannot represent a nonzero element. The general pair survives.

The factorization exists for every element of the null cone, including the origin. The four-factor refinement is the privilege of the N-regular elements, where the modulus is nonzero.

Summary of Notation

Further Reading

• Roger Penrose, "A generalized inverse for matrices", Proceedings of the Cambridge Philosophical Society 51 (1955) 406–413. The four equations and the generalized inverse.
• Gene H. Golub and Charles F. Van Loan, Matrix Computations, 4th edition (Johns Hopkins University Press, 2013). The singular value decomposition, the Moore–Penrose inverse and the rank-one limits.
• Nicholas J. Higham, Functions of Matrices: Theory and Computation (SIAM, 2008). The matrix square root and the polar factor at a rank drop.
• Max-Albert Knus, Alexander Merkurjev, Markus Rost and Jean-Pierre Tignol, The Book of Involutions, Colloquium Publications 44 (American Mathematical Society, 1998). Algebras with involution, the Hermitian cone and the partial isometries of a rank-one factor.