Diagonal Perturbations of Linear Equations

In summary, there is no existing theorem for characterizing how the components of the solution x change as the elements of D change, as the form of A+D is arbitrary due to lack of information about A. However, a blockwise addition and inversion can be performed to establish a system of differential equations when the entries of D vary, but each new size of M creates a new problem.
  • #1
JohnSimpson
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Consider the system of linear equations (A+D)x=b, where D is a positive semidefinite diagonal matrix. Assume for simplicity that (A+D) is full rank for any D that we care about. In my particular case of interest, D has the form D = blkdiag(0,M) for some positive diagonal matrix M. So, a subset of the diagonal entries of A are being perturbed.

Are there any off-the-shelf theorems that characterize how the components of the solution x change as the elements of D change? Intuitively as M increases in size, the bottom elements of x should shrink. Coupling through off-diagonal elements of A should then also shrink the top elements a little bit.

Thanks
-John
 
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  • #2
No, there isn't such a thing, simply because we have no information about ##A##, which makes the form of ##A+D## arbitrary. You could only perform a blockwise addition, possibly an inversion, and establish a system of differential equations, if the entries of ##D## variate. But any new size of ##M## gives you a new problem.
 

FAQ: Diagonal Perturbations of Linear Equations

What are diagonal perturbations of linear equations?

Diagonal perturbations of linear equations refer to small changes or variations made to the coefficients of a linear system of equations. These changes are made to the diagonal elements of the coefficient matrix, hence the term "diagonal perturbations".

Why are diagonal perturbations important in scientific research?

Diagonal perturbations are important because they allow scientists to study the effects of small changes in a linear system. This can help in understanding the stability and behavior of the system, and can also provide insights into the sensitivity of the system to these changes.

How are diagonal perturbations applied in real-world problems?

Diagonal perturbations are commonly used in various fields such as engineering, physics, and economics to model and analyze real-world problems. They can be used to study the effects of uncertainties or errors in measurements, as well as to analyze the robustness of solutions to a given system.

Can diagonal perturbations affect the solutions of a linear system?

Yes, diagonal perturbations can significantly affect the solutions of a linear system. Even small changes in the diagonal elements of the coefficient matrix can result in different solutions, and in some cases, may even render the system unsolvable.

How are diagonal perturbations different from other types of perturbations?

Diagonal perturbations differ from other types of perturbations, such as off-diagonal perturbations or additive perturbations, in that they only affect the diagonal elements of the coefficient matrix. This makes them easier to analyze and understand, and allows for more precise control over the changes being made to the system.

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