Solve matrix equation $X = I + A + AXA$

Question

Notations

$X$ : p times p positive semi definite, unknown.

$I$ : p times p identity matrix, known.

$A$ : p times p positive semi definite, known.

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I want to find X, but it is difficult to me.

Thus, instead of exact solution, I used following iterative method to find the approximate solution.

Step 1) : initialize $X(0)$

Step 2) : $X(k) = I + A + A X(k-1) A$

Step 3) : until $|X(k) - X(k-1)|_F < tol$

But, this method highly depends on the initial value X(0) and is vulnerable to diverge.

Please modify my method or propose another method.

Regard, Han

Answer 1

I suggest you to follow this for a strict justification that $X=\left(I-A\right)^{-1}$.

First it is easy to see that each eigenvalue $\lambda$ of $A$ satisfies $0\le\lambda <1$ for the given equation to have a solution. To illustrate, let $Av = \lambda v$ for some $v\in\mathbb{R}^p\setminus\left\{0\right\}$. Then the given equation yields \begin{align*} v^t Xv &= v^t v + v^t Av + v^t AXAv \\&= v^t v + \lambda v^t v + \lambda^2 v^t Xv, \end{align*} or, \begin{align*} \left(1-\lambda^2 \right)v^t Xv = \left(1+\lambda\right)v^t v. \end{align*} If $\lambda\ge1$, as $v^t v>0$, the right hand side must be positive. While the positive semidefiniteness of $X$ gives $v^t Xv \ge0$ and hence the left hand side is $\le 0$, yielding a contradiction. This together with the positive semidefiniteness of $A$ proves the claim.

Now multiplying $\left(I-A\right)$ to the given equation by left, it becomes $$ \left(I-A\right)X = I-A^2 + A \left(I-A\right)XA $$ or we can write $$ I-\left(I-A\right)X = A\left(I-\left(I-A\right)X\right)A. $$ Replacing $I-\left(I-A\right)X$ with $Y$, we arrive at a simpler form $$ Y=AYA. $$ Let $v$ and $w$ be eigenvectors of $A$ corresponding to the eigenvalues $\lambda$ and $\mu$. Then \begin{align*} v^t Y w = v^t AYA w = \lambda\mu\left( v^t Y w\right) \end{align*} holds, which together with the fact that $0\le\lambda, \mu<1$ force $v^t Y w =0$. Note, however, that $A$ is symmetric and thus is diagonalizable, that is, its eigenvectors span any vectors in $\mathbb{R}^p$. This implies $v^t Y w =0$ for any $v,w\in\mathbb{R}^p$ which holds only if $Y=O$. From this we have $\left(I-A\right)X = I$ and therefore $X=\left(I-A\right)^{-1}$.

Answer 2

From induction we have $$X=I+A+\cdots +A^{2n-1}+A^nXA^n$$by tending $n\to \infty$ and defining $B=\sum_{n=0}^\infty A^n$, we have $B=(I-A)^{-1}$ only if $I-A$ is invertible, unless $X$ does not exist. Also all the eigenvalues of $A$ must have $|\cdot |<1$. Therefore $A^n\to 0$ and $$X=(I-A)^{-1}$$

Answer 3

More generally, let $K$ be an algebraically closed field and $A\in M_n(K)$ with $spectrum(A)=(\lambda_i)_i$.

We consider the equation in $X\in M_n(K)$: $(*)$ $X=I+A+AXA$.

$\textbf{Proposition}$. If for every $(i,j),$ $\lambda_i\lambda_j\not= 1$ then $(*)$ has the sole solution

$X=(I-A)^{-1}$.

$\textbf{Proof}$. We use the Ramanasa's trick. If $Y=I-(I-A)X$, then $f(Y)=AYA-Y=0$. $f$ is linear and $spectrum(f)=(\lambda_i\lambda_j-1)_{i,j}$.

(cf. https://en.wikipedia.org/wiki/Kronecker_product).

Thus $0\notin spectrum(f)$, $f$ is one to one and $Y=0$. Since $1\notin spectrum(A)$, we deduce that $X=(I-A)^{-1}$. $\square$