Matrix sign function

The Newton iteration can be derived by observing that ${\displaystyle \operatorname {csgn} (x)={\sqrt {x^{2}}}/x}$, which in terms of matrices can be written as ${\displaystyle \operatorname {csgn} (A)=A^{-1}{\sqrt {A^{2}}}}$, where we use the matrix square root. If we apply the Babylonian method to compute the square root of the matrix ${\displaystyle A^{2}}$, that is, the iteration ${\textstyle X_{k+1}={\frac {1}{2}}\left(X_{k}+AX_{k}^{-1}\right)}$, and define the new iterate ${\displaystyle Z_{k}=A^{-1}X_{k}}$, we arrive at the iteration

${\displaystyle Z_{k+1}={\frac {1}{2}}\left(Z_{k}+Z_{k}^{-1}\right)}$,

where typically ${\displaystyle Z_{0}=A}$. Convergence is global, and locally it is quadratic.[1][2]

The Newton iteration uses the explicit inverse of the iterates ${\displaystyle Z_{k}}$.

To avoid the need of an explicit inverse used in the Newton iteration, the inverse can be approximated with one step of the Newton iteration for the inverse, ${\displaystyle Z_{k}^{-1}\approx Z_{k}\left(2I-Z_{k}^{2}\right)}$, derived by Schulz(de) in 1933.[4] Substituting this approximation into the previous method, the new method becomes

${\displaystyle Z_{k+1}={\frac {1}{2}}Z_{k}\left(3I-Z_{k}^{2}\right)}$.

Convergence is (still) quadratic, but only local (guaranteed for ${\displaystyle \|I-A^{2}\|<1}$).[1]

Theorem:[2][3] Let ${\displaystyle A,B,C\in \mathbb {R} ^{n\times n}}$ and assume that ${\displaystyle A}$ and ${\displaystyle B}$ are stable, then the unique solution to the Sylvester equation, ${\displaystyle AX+XB=C}$, is given by ${\displaystyle X}$ such that

${\displaystyle {\begin{bmatrix}-I&2X\\0&I\end{bmatrix}}=\operatorname {csgn} \left({\begin{bmatrix}A&-C\\0&-B\end{bmatrix}}\right).}$

Proof sketch: The result follows from the similarity transform

${\displaystyle {\begin{bmatrix}A&-C\\0&-B\end{bmatrix}}={\begin{bmatrix}I&X\\0&I\end{bmatrix}}{\begin{bmatrix}A&0\\0&-B\end{bmatrix}}{\begin{bmatrix}I&X\\0&I\end{bmatrix}}^{-1},}$

since

${\displaystyle \operatorname {csgn} \left({\begin{bmatrix}A&-C\\0&-B\end{bmatrix}}\right)={\begin{bmatrix}I&X\\0&I\end{bmatrix}}{\begin{bmatrix}I&0\\0&-I\end{bmatrix}}{\begin{bmatrix}I&-X\\0&I\end{bmatrix}},}$

due to the stability of ${\displaystyle A}$ and ${\displaystyle B}$.

The theorem is, naturally, also applicable to the Lyapunov equation. However, due to the structure the Newton iteration simplifies to only involving inverses of ${\displaystyle A}$ and ${\displaystyle A^{T}}$.

There is a similar result applicable to the algebraic Riccati equation, ${\displaystyle A^{H}P+PA-PFP+Q=0}$.[1][2] Define ${\displaystyle V,W\in \mathbb {C} ^{2n\times n}}$ as

${\displaystyle {\begin{bmatrix}V&W\end{bmatrix}}=\operatorname {csgn} \left({\begin{bmatrix}A^{H}&Q\\F&-A\end{bmatrix}}\right)-{\begin{bmatrix}I&0\\0&I\end{bmatrix}}.}$

Under the assumption that ${\displaystyle F,Q\in \mathbb {C} ^{n\times n}}$ are Hermitian and there exists a unique stabilizing solution, in the sense that ${\displaystyle A-FP}$ is stable, that solution is given by the over-determined, but consistent, linear system

${\displaystyle VP=-W.}$

Proof sketch: The similarity transform

${\displaystyle {\begin{bmatrix}A^{H}&Q\\F&-A\end{bmatrix}}={\begin{bmatrix}P&-I\\I&0\end{bmatrix}}{\begin{bmatrix}(-A-FP)&-F\\0&(A-FP)\end{bmatrix}}{\begin{bmatrix}P&-I\\I&0\end{bmatrix}}^{-1},}$

and the stability of ${\displaystyle A-FP}$ implies that

${\displaystyle \left(\operatorname {csgn} \left({\begin{bmatrix}A^{H}&Q\\F&-A\end{bmatrix}}\right)-{\begin{bmatrix}I&0\\0&I\end{bmatrix}}\right){\begin{bmatrix}X&-I\\I&0\end{bmatrix}}={\begin{bmatrix}X&-I\\I&0\end{bmatrix}}{\begin{bmatrix}0&Y\\0&-2I\end{bmatrix}},}$

for some matrix ${\displaystyle Y\in \mathbb {C} ^{n\times n}}$.

The Denman–Beavers iteration for the square root of a matrix can be derived from the Newton iteration for the matrix sign function by noticing that ${\displaystyle A-PIP=0}$ is a degenerate algebraic Riccati equation[3] and by definition a solution ${\displaystyle P}$ is the square root of ${\displaystyle A}$.