Motivation

Solution of (large) linear systems

• $A\in\mathbb{M}_n(\mathbb{C})$ and $x,b\in \mathbb{C}^n$
• $A$ is large non-Hermitian positive definite

Let $A = H + S$, where $H=\frac{A+A^*}{2}$ and $S=\frac{A-A^*}{2}$

HSS algorithm (Bai, Golub and Ng, 2003)

HSS algortihm

• Unconditionally convergent to the solution of $Ax=b$, for any $\alpha>0$
• Upper bound of the contraction factor depends on $\sigma(H)$ but not on $\sigma(S)$
• Optimal $\alpha$ is given by $\alpha_{opt}:=\sqrt{\lambda_{max}(H)\lambda_{min}(H)}$
• The two-step iteration is solved via some CG + Krylov subspace method (IHSS)
• Efficient solver for saddle-point problems for discretized elliptic PDEs (Benzi, Gander and Golub, 2003)

Skew shifted HSS algortihm (Greif, 2022)

• $J$ is the standard symplectic matrix
• Not unconditionally convergent to the solution of $Ax=b$, but there is a critical value for $\alpha$ depending on the eigenvalues of the skew-Hamiltonian matrix $JS$

or

(1)

(2)

Skew shifted HSS algortihm (Greif, 2022)

Rmk. The scheme (1) is equivalent to

which corresponds to the splitting of $JA$ in the Hamiltonian and skew-Hamiltonian parts $JH$ and $JS$

Rmk. Hermitian and skew-Hermitian and skew-Hamiltonian and Hamiltonian are of Jordan and Lie algebras pairs

Q. How the Lie-Jordan theory is linked to the HSS scheme? 

Lie and Jordan algebras

Def. A Lie algebra $\mathfrak{g}$ is a vector space with a bilinear operator $[\cdot,\cdot]:\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$, such that for all $A,B,C\in\mathfrak{g}$:

• $[[A,B],C] + [[B,C],A] + [[C,A],B] = 0$ (Jacobi)
• $[A,B] = - [B,A]$

Def. A Jordan algebra $(\mathbb{J},\circ)$ is a non-associative algebra over a field, such that for all $A,B\in \mathbb{J}$:

• $A\circ B = B\circ A$
• $(A\circ B)\circ A^2 = A\circ ( B\circ A^2)$

Es. $\mathbb{M}_n(\mathbb{R})$ is both a Lie and Jordan algebra, w.r.t. to the commutator and anti-commutator

Lie-Jordan splitting

Key ingredient: any $A\in\mathbb{M}_n(\mathbb{R})$ can be uniquely decomposed in a symmetric and a skew-symmetric part

$\mathbb{M}_n(\mathbb{R})\cong\mathfrak{so}_n(\mathbb{R}) \oplus sym_n(\mathbb{R})$

• $\mathfrak{so}_n(\mathbb{R})$ is a real Lie algebra w.r.t. the matrix commutator $[A,B]=AB-BA$
• $sym_n(\mathbb{R})$ is a real Jordan algebra w.r.t. the matrix anticommutator $A\circ B = AB+BA$
• The splitting above is known as the Polar decomposition at the algebra level or the Cartan decomposition w.r.t. the involution $\cdot^T$

Lie-Jordan splitting

More generally, given any $J\in\mathbb{M}_n(\mathbb{R})$ invertible symmetric or skew-symmetric, we get a pair of quadratic Lie and Jordan algebras

where
$\mathfrak{g}_J:=\lbrace A\in\mathbb{M}_n(\mathbb{R})$ such that $A^TJ + JA=0\rbrace$

$\mathbb{J}:=\lbrace A\in\mathbb{M}_n(\mathbb{R})$ such that $A^TJ - JA=0\rbrace$

$\mathbb{M}_n(\mathbb{R})\cong\mathfrak{g}_J \oplus_S \mathbb{J}$

• The splitting is orthogonal w.r.t.: $<A,B>_S :=Tr(A^TSB)$, where $S=JJ^T$

J-HSS scheme 

Let $A = H + S$, where $H=\frac{J^{-1}(A^TJ+JA)}{2}$ and $S=\frac{J^{-1}(A^TJ-JA)}{2}$

Thm.

• Let $J$ be symmetric. Then, the scheme above converges toward the solution of $Ax=b$ if $max_{\lambda\in\sigma(JH)}|\frac{\alpha-\lambda}{\alpha + \lambda}|<1$
• Let $J$ be skew-symmetric. Then, the scheme above converges toward the solution of $Ax=b$ if $max_{\lambda\in\sigma(JS)}|\frac{\alpha-\lambda}{\alpha + \lambda}|<1$

J-HSS scheme 

• By the Sylvester's inertia theorem, when $J$ is congruent to a diagonal matrix with zeros and $\pm 1$ equal to the rank of $J$, whereas when J is skew is a block diagonal with zeros and symplectic blocks equal to half the rank of $J$. Hence, in this case the Lie algebra g is isomorphic to $\mathfrak{u}(p,q)$ or $\mathfrak{sp}(n)$.
• $J$ neither symmetric or skew can be reconducted to the previous case
• Considering $JAx=Jb$, brings us to the classical HSS scheme

Q. Are there Lie-Jordan splitting not isomorphic to the Hermitian and skew-Hermitian one?

Formally real Jordan algebras

In 1934, Jordan, Von Neumann and Wigner classified the simple formally real Jordan algebras, i.e. real Jordan algebras such that:

if $A_1^2+A_2^2+...+A_n^2=0$, then $A_1=A_2=...=A_n=0$

They are of three kinds:

1. Hermitian matrices for any $n\geq 1$ with coefficients in $\mathbb{R},\mathbb{C},\mathbb{H}$
2. $JSpin_n$: $\mathbb{R}\textbf{1}\oplus\mathbb{R}^n$ with $(a\textbf{1},\textbf{v})\circ(b\textbf{1},\textbf{w})=((ab+<\textbf{v},\textbf{w}>)\textbf{1},a\textbf{v}+b\textbf{w})$
3. Hermitian matrices for $n=3$ with coefficients in $\mathbb{O}$ [Albert algebra]

Rmk.

• $JSpin_n$ can be embedded in $sym_{2^n}(\mathbb{R})$
• The Albert algebra is the only formally real exceptional algebra, since it cannot be embedded in a associative algebra

Formally real Jordan algebras

• The Hermitian and skew-Hermitian splitting and the relative HSS scheme can be extended to matrix with coefficients in $\mathbb{R},\mathbb{C},\mathbb{H}$
   
• $\mathbb{M}_n(\mathbb{H})\cong\mathbb{M}_{2n}(\mathbb{C})$, hence all the analysis for quaternionic matrices can be performed using complex matrices
   
• A $JSpin_n$ Jordan algebra has a Lie algebra complement $\mathfrak{g}\subset\mathfrak{so}(2^n)$
   
• The Albert algebra is not clearly treatable, since it cannot be represented via real or complex matrix calculus

Freudenthal-Tits magic square

• Another way to associate a Lie algebra to a Jordan algebra $\mathbb{J}$ is to define the structure algebra $str(\mathbb{J})=\mathbb{J}\oplus Der(\mathbb{J})$.
  Es. $\mathbb{J}=sym_n(\mathbb{R})$ has $Der(\mathbb{J})\cong\mathfrak{so}(n)$ and $str(\mathbb{J})\cong\mathbb{M}_n(\mathbb{R})$
   
• The Freudenthal-Tits construction $FT (C, J) := Der(C)⊕(C0⊗J0)⊕Der(J)$ links $3\times 3$ Hermitian matrices to semisimple Lie algebras

Conclusions and future work

• The HSS scheme can be directly extended to J-quadratic Lie and Jordan algebras
• The HSS scheme works for Hermitian matrices with coefficients in $\mathbb{R},\mathbb{C},\mathbb{H}$
• Is there any case in which is convenient to consider the $JSpin_n$ structure and its Lie algebra complement?
• The exceptional cases emerge while considering octonions, but only for dimension 3, which makes the interest for iterative schemes purely academic
• Extend the analysis of the HSS algorithm for any real Jordan algebra of matrices, considering the structure algebra $str(\mathbb{J})=\mathbb{J}\oplus Der(\mathbb{J})$