# Reconstructing a Matrix in the $\Bbb{R}^3$ space with $3$ eigenvalues, from $3$ matrices in $\Bbb{R}^2$ or $\Bbb{ C}^2$ space with $2$ eigenvalues

I have a matrix which represents a closed loop matrix of a control system with delays (Control Systems Theory) in $\Bbb{R}^3$ space that has $3$ eigenvalues. Through some process I have obtained three different matrices in $\Bbb{R}^2$ and sometimes in $\Bbb{C}^2$ which represent a part of the control system and each matrix has just two of the three original eigenvalues.

What I want is to try to reconstruct the original $\Bbb{R}^3$ matrix. I do not know which operation or algebraic manipulation should I do to get the $\Bbb{R}^3$ system from two of the three systems obtained.

To show a numerical example we have the following:

$$A=\begin{pmatrix}05099 & 0.2649 & 0.01\\ -0.99 & -0.485 & 0.01\\ -1 & -1.5 & 0\end{pmatrix}$$

It has the following eigenvalues: $-0.0875+0.2592i$, $-0.0875-0.2592i$ and $0.1998$. Through some process I have obtained three different systems:

$$A_{11}=\begin{pmatrix}0.5825-0.1494i & 0.2796-0.1092i\\ -0.9170-0.1502i & -0.4702-0.1097i\end{pmatrix}$$

Its eigenvalues are: $0.1998$ and $-0.0875-0.2592i$

$$A_{22}=\begin{pmatrix}0.5825+0.1494i & 0.2796+0.1092i\\ -0.9170+0.1502i & -0.4702+0.1097i\end{pmatrix}$$

Its eigenvalues are: $0.1998$ and $-0.0875+0.2592i$

$$A_{33}=\begin{pmatrix}0.3502 & 0.2249\\ -1.1505 & -0.5252\end{pmatrix}$$

Its eigenvalues are: $-0.0875-0.2592i$ and $-0.0875+0.2592i$

Thanks for your help. I hope it is clear, but if not please ask.

Gina Torres

Answers to the question about the process that i have done.

Yes i have missed the i's, sorry for that, the numbers are complex. The process what i have done is the following, this matrix A, named extended in the language that i am using contains n+1 eigenvalues (3 eigenvalues in the case that i am talking about). Is possible to obtain these three new models through this $A_{11}=SAMP^kS^T(SMP^kS^T)^{-1}$ which is the same $A_{11}=SMEP^kS^T(SMP^kS')^{-1}$ where S is a selection matrix, in this case is the following $$S=\begin{pmatrix}1 & 0 & 0\\0 & 1 & 0\end{pmatrix}$$ and P is a permutation Matrix $$P=\begin{pmatrix}0 & 0 & 1\\ 1 & 0 & 0\\0 & 1 & 0\end{pmatrix}$$ for eigenvectors. E is a diagonal Matrix which contains the eigenvalues and M is the eigenvectors matrix. Another way to express that solutions is $A=(SMP^{k}S')S(MP^k)^{-1}A(MP^k)S'(SMP^kS')^{-1}$. What i have done is like a projection from $R^3$ to $R^2$ or $C^2$ eliminating one eigenvalue in each case to obtain A11, A22, A33.

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The $\mathrm i$s are missing in the eigenvalues. – joriki Dec 10 '12 at 14:19
Are there some $i$s missing? Where you've written a real number as a sum of two real numbers... This isn't my area, so maybe you're using suggestive standard notation, but I'd be slightly surprised if anybody could help without more details as to what "some process" is. – Matthew Pressland Dec 10 '12 at 14:19
What is the process you have employed to get $A_{11},A_{22},A_{33}$ from $A$? Without knowing what you have done, reversing it will be difficult – dexter04 Dec 10 '12 at 14:20
The complete explanation is in the question that i have edited some minutes ago. – Gina Torres Dec 10 '12 at 15:03