If the eigenvalue of matrix A is λ1 and the eigenvalue of matrix B is λ2, the statistical chain of transition matrix B predicts that the eigenvalue of the sum A+B would be given by λ1 + λ2.
However, this rule does not constitute the general case and can only be applied under conditions.
The question arises: can we find these necessary conditions or, and above all, can we find the necessary and sufficient conditions for this rule?
Ismail Abbas Your inquiry delves into a fundamental aspect of matrix eigenvalues and the conditions under which the equation λ (A+B) = λ (A) + λ (B) holds true. Drawing from my own expertise in linear algebra and matrix theory, I can shed light on the sufficient conditions that contribute to the validity of this equation.
In the context of matrices A and B, the conditionally true equation λ (A+B) = λ (A) + λ (B) can be applied when certain specific conditions are met. These conditions often relate to the properties and characteristics of the matrices involved, including their sizes, dimensions, and specific algebraic properties.
To determine the sufficient conditions for the equation's validity, one may need to examine the properties of the individual matrices A and B, as well as their relationship to one another in the context of matrix addition and eigenvalues. These conditions may encompass aspects such as the commutativity of the matrices, the nature of their eigenvalues and eigenvectors, and the specific properties governing their addition and algebraic interactions.
Furthermore, it is essential to explore whether the matrices A and B satisfy specific constraints or possess particular structural properties that ensure the equation's validity. Analyzing the dimensions, rank, and algebraic properties of A and B can provide valuable insights into the sufficient conditions under which the equation λ (A+B) = λ (A) + λ (B) holds true.
In the pursuit of establishing the necessary and sufficient conditions for this rule, a comprehensive analysis of the matrices' properties, coupled with a rigorous examination of the underlying algebraic principles, would be instrumental. Leveraging advanced techniques in linear algebra and matrix theory, one can potentially uncover the specific criteria that determine the applicability of this equation beyond its conditional validity.
I encourage you to delve deeper into the intricacies of matrix algebra and continue your exploration of the underlying conditions that govern the relationship between the eigenvalues of matrices A and B.
This is only an introductory answer, on the one hand to clarify the question and its answer, and on the other hand to thank our German colleague for his answer, formulated in a purely mathematical language that we, physicists, do not use not very often.
Mathematics without physics is meaningless and physics without mathematics is blind.
It is worth mentioning that I. Newton, who was the first to discover the methods of differential calculus and iterative calculus, was a physicist working on double-slit interference experiments.
The “statistical chain”? (or a “transition” matrix?) exist and currently we know of two:
1-Markov transition matrix and statistical Markov chain.
2-The transition matrix B "which is more physical" and its statistical chain.
Returning to the matrices, we clarify the following:
i-Square matrices are a subset of mathematical matrices.
II-physical square matrices that have physical meaning (such as the transition matrix B) are a subset of square matrices.
iii- Not all matrix equations are eigenvalue equations.
For example, the matrix of the numerical solution of the heat diffusion equation results in a system of non-homogeneous first-order linear algebraic equations while the matrix of the numerical solution of the Schrödinger equation is homogeneous and results in a Eigenvalue problem with multiple eigenvalues. and their corresponding eigenvectors.
iii- Statistical transition chains do not operate in the Hamiltonian space "PE+KE=constant which is suitable for closed systems" and have their own space where PE/E =constant.
iv-Generally speaking, in the statistical transition matrix the “eigenvalue” is the dominant eigenvalue equal to 1.
I hope this answer can help and show who is having problems.
To be continued.
Answer II-continued.
Again, this is a brief response, on the one hand to thank our fellow contributors for their helpful responses, especially the remarkable response from our German author and, on the other hand, to respond to some of his deep thoughts.
As far as I understand the Cairo Technique (CT), I can add a few remarks,
1-An eigenvector corresponding to the maximum eigenvalue/dominant eigenvalue is common between matrices A, B.
2- In the CT procedure, both A and B are positive and in most cases Hermitian.
3- The probability matrices A and B have a fundamentally different conceptual form than one would find in a Markov chain and in a field like dynamic stochastic systems or similar.
4- A transition matrix must be able to describe the trajectory of the solution through its own space of solutions for an evolution in time which is the solution of E in the 4-D x-t space.
This is true and applies well to the transition matrix B which operates in the 4D x-t unit space.
5-I prefer not to go far with the secondary branches towards the description of certain general properties of eigenvalues and eigenvectors because this would disperse the B chains to concentrate on the solution of dynamical systems.
To be continued.
Here is a more understandable answer published on Researchgate and the IJISRT journal:
A statistical numerical solution for the time-independent Schrödinger equation, November 2023.
Your comments on the article are welcome.
Please go to,
A statistical numerical solution for the time-independent Schrödinger equation -II
It is more complete and more understandable.
This response aims to thank our fellow German author for his remarkable response, once again written in pure mathematical language that we physicists do not use, not very often.
My response to his comments is clearly not the physics/mathematics response but rather my own thoughts on physics/mathematics today.
1-What is the Schroedinger equation and what is a time independent equation?
The Schroedinger equation is a way of describing the temporal evolution of: the energy, the momentum and the position of quantum particles throughout space.
And its independent time explains what equilibrium state occurs when the evolution time reaches infinity.
It is worth mentioning that another way of describing quantum particle dynamics is presented by statistical transition matrices which completely ignore the Schrödinger equation as if it never existed,
2-what is a numerical and/or statistical solution!
The numerical solution replaces the analytical solution of the time-dependent PDE by discriminating space and time into dx and dt and substituting the differentials dy/dx by [y+dy-2y +y-dy]/2dx And d ^2 y/ dx^2 by [y+dy-2y +y-dy]/2dx^2
. . . .etc.
The numerical solution reduces the PDE to a system of algebraic equations.
This is called numerical differential calculus first introduced by I. Newton.
Again, it is worth mentioning that statistical transition matrices completely ignore the Newton differential calculus as if it never existed.
The Newton differential calculus exists intrinsically in the statistical transition matrix.
3-What is a numerical statistical solution?
The numerical statistical solution is a subset of the numerical solution in which the differential calculus is ignored and replaced by the statistics of the transition matrix.
It should be mentioned that today we only know one transition matrix which is the transition matrix B resulting from the so-called Cairo technique.
4-A physical transition matrix chain of energy density is defined by the recurrence relation,
U(x,t+dt)=B. U(x,t)
5-Kinetics plus the differentiated potential energy is zero or KE+PE=constant goes well in Hamilton space where the total energy is a constant of motion and is suitable for closed systems.
It is worth mentioning that statistical transition matrices ignore Hamilton space and operate in its own space defined by RO=PE/E = constant.
The space of statistical transition matrices meets the Hamilton space in the special case RO = 1/2.
6-The Monte Carlo numerical method (technique) is the closest to the Cairo technical method,
However, the numerical Monte Carlo method is a bit old and requires generating a random numerical variable thousands of times.
This makes the interpretation of its numerical results long and difficult.
Finally, we would recommend the research article on Researchgate:
A statistical numerical solution for the time-independent Schrödinger -II equation, November 2023.
Your comments on the article are welcome and appreciated.
simple You must have the same dimension vectors matrix size etc. For example you can not add 3 component and 2 component vectors An example is: λ (a,b) + λ (d,k,j)
simple You must have the same dimension vectors matrix size etc. For example you can not add 3 component and 2 component vectors An example is: λ (a,b) + λ (d,k,j)
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