Find $\mathrm{tr}(AX^{-1}A'X)$

Given $$A=\begin{bmatrix} a_1 & & \\ & a_2 & \\ &&a_2 \end{bmatrix}$$ and $$B\in\mathbb{R}^{3\times2}$$, where $$a_1,a_2>1$$, and $$X=AX(I+BB'X)^{-1}A$$.

1) Find $$\mathrm{tr}(AX^{-1}AX)$$ in terms of $$a_i$$.

2) Show that $$\frac{x_1\cdot\begin{vmatrix} x_2 & x_{23}\\ x_{23}&x_3 \end{vmatrix}}{|X|}$$ is independent of $$B$$, where $$|X|=\mathrm{det}X$$.

My attempt: $$X$$ is the solution to discrete-time algebraic Riccati equation.

Let $$X=\begin{bmatrix} x_1 & x_{12} & x_{13}\\ x_{12} & x_2 & x_{23}\\ x_{13}&x_{23}&x_3 \end{bmatrix}$$ and $$z=x_1\cdot\begin{vmatrix} x_2 & x_{23}\\ x_{23}&x_3 \end{vmatrix}$$, then I found that

\begin{align} \mathrm{tr}(AX^{-1}AX) =(a_1-a_2)^2\frac{z}{|X|}+(2a_1a_2+a_2^2). \end{align}

I did simulation by fixing $$A$$ and found out that $$\frac{z}{|X|}$$ doesn't depend on $$B$$, thus $$\mathrm{tr}(AX^{-1}AX)$$ also doesn't depend on $$B$$ (as long as $$B$$ is full rank).

Observation 1: $$A=\begin{bmatrix} 7 & & \\ & 2 & \\ &&2 \end{bmatrix}$$ and $$A=\begin{bmatrix} 2 & & \\ & 7 & \\ &&7 \end{bmatrix}$$ has equal $$\frac{z}{|X|}$$.

Observation 2: The larger the gap between $$a_1$$ and $$a_2$$, the smaller $$\frac{z}{|X|}$$ and vice-versa.

• what are $B'$ and $A'$? Jun 4, 2020 at 17:35
• @mrcntn Matrix transpose
– Lee
Jun 4, 2020 at 17:56
• Forgive me if I am mistaken, but isn't $\text{tr}(AX^{-1}A'X) = \text{tr}(A'A)$ since the matrices inside the trace can commute? Jun 4, 2020 at 18:10
• @Theorem They cannot commute like that, they should preserve the order, $\mathrm{tr}(AX^{-1}A'X)=\mathrm{tr}(X^{-1}A'XA)=\mathrm{tr}(A'XAX^{-1})=\mathrm{tr}(XAX^{-1}A')$
– Lee
Jun 4, 2020 at 18:19

Assume that $$\det X \ne 0$$.
From $$X = AX(I + BB^\mathsf{T}X)^{-1}A$$, we have $$I + BB^\mathsf{T}X = AX^{-1}AX$$ and $$BB^\mathsf{T} = AX^{-1}A - X^{-1}$$. Let $$Y = X^{-1}$$. We have $$BB^\mathsf{T} = AYA - Y$$. Let $$Y=\begin{bmatrix} y_1 & y_{12} & y_{13}\\ y_{12} & y_2 & y_{23}\\ y_{13}&y_{23}&y_3 \end{bmatrix}, \quad B = \begin{bmatrix} b_1 & b_2 \\ b_3 & b_4 \\ b_5 & b_6 \end{bmatrix}.$$ From $$BB^\mathsf{T} = AYA - Y$$, we solve $$Y$$ uniquely \begin{align} y_1 &= \frac{b_1^2 + b_2^2}{a_1^2 - 1}, \quad y_{12} = \frac{b_1b_3 + b_2b_4}{a_1a_2 - 1}, \quad y_{13} = \frac{b_1b_5 + b_2 b_6}{a_1a_2 - 1}, \\ y_2 &= \frac{b_3^2 + b_4^2}{a_2^2 - 1}, \quad y_{23} = \frac{b_3b_5 + b_4b_6}{a_2^2-1}, \quad y_3 = \frac{b_5^2+b_6^2}{a_2^2-1}. \end{align}
By using the well-known relation $$M^{-1} = \frac{1}{\det M} \mathrm{adj}(M)$$, from $$Y = X^{-1}$$ and $$X = Y^{-1}$$, we have \begin{align} y_1 &= \frac{\begin{vmatrix} x_2 & x_{23}\\ x_{23}&x_3 \end{vmatrix}}{\det X}, \\ x_1 &= \frac{\begin{vmatrix} y_2 & y_{23}\\ y_{23}&y_3 \end{vmatrix}}{\det Y}. \end{align} Thus, we have $$\frac{z}{\det X} = y_1x_1 = y_1 \frac{\begin{vmatrix} y_2 & y_{23}\\ y_{23}&y_3 \end{vmatrix}}{\det Y} = \frac{(a_1a_2-1)^2}{(a_1-a_2)^2}.$$ Then, we have $$\mathrm{Tr}(AX^{-1}AX) = (a_1-a_2)^2\frac{z}{\det X}+(2a_1a_2+a_2^2) = a_1^2a_2^2 + a_2^2 + 1.$$