A very interesting application of matrix theory and linear algebra is on mathematical analysis, formulation and studies on Quantum information and Quantum communications.
Dear Debopam Ghosh
Could you please give me some recent references?
Thank you in advance
Dear Abdelaziz Mennouni ,
Matrices are essential to study all scientific topics.
Their properties provide a robust tool to describe almost all mathematical models. For example:
Solving linear systems, approximating the roots (eigenvalues) of polynomials, linearizing non-linear systems, dynamical systems, Jacobian, Hessian, Control theory, operators, probability theory, algebraic structures, Fourier and Laplace transforms, optimization problems, all branches in applied physics, graph theory, number theory, economic, differential topology and endless disciplines in pure and applied sciences.
So, any advances and discoveries in the matrix theory will reflect positively on all the mentioned topics.
One of the famous open problems in matrix theory is the following:
If A and B are two square non- commuting matrices of the same dimension. If we know the eigenvalues of both matrices A and B.
What are the eigenvalues of A + B?
There are some partial results, as in the paper:
https://www.ams.org/notices/200102/fea-knutson.pdf,
but the general case is still one of the most challenging problems.
Best regards
A5. Extensions from fields to rings.
Answer 5, posted on May 24, 2019.
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A very useful subject of research in matrix theory
is the extensions of basic major theorems
from matrices with entries in a field
to matrices with entries in a ring.
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It is important, because in physics and engineering
it is exceptional that the entries of matrices are constant.
Most of the time, the entries are functions of variables,
so that the entries belong to
a ring of continuous functions,
or a ring of differentiable functions,
or a ring of smooth functions,
or a ring of analytic functions,
or …
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With best regards, Jean-Claude
Dear Jean-Claude
Thank you so much.
Could you please give me some recent references on this subject of research ?
Best regards
Dear Issam Kaddoura
Thank you very much for your help and support.
Best regards,
Abdelaziz Mennouni
A8. Similarities of matrix-valued functions.
Answer 8, posted on May 25, 2019.
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To answer the question posted by Abdelaziz Mennouni
in answer 6, I can give the reference to my following
research project:
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Similarities and equivalences of matrix-valued functions
https://www.researchgate.net/project/Similarities-and-equivalences-of-matrix-valued-functions
Click on “Show details” on the seventh line.
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As for any research project, it is terribly time consuming
to find the most important references on the subject.
The information on the Internet is posted in chaos form.
Try to make a search with “Matrices over rings”.
You will see a chaos of information.
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With best regards, Jean-Claude
A9. Geometric mean for matrices
Answer 9, posted on May 27, 2019.
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I found the following major references
thanks to the following question:
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Geometric mean of more than 2 SPD matrices?
https://www.researchgate.net/post/Geometric_mean_of_more_than_2_SPD_matrices
posted by Hadi Tabealhojeh
https://www.researchgate.net/profile/Hadi_Tabealhojeh
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and thanks much more to answer 1
posted by Issam Kaddoura
https://www.researchgate.net/profile/Issam_Kaddoura
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Here is more information.
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The reference is:
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A Differential Geometric Approach to the Geometric Mean
of Symmetric Positive-Definite Matrices
Maher Moakher
SIAM J. Matrix Anal. Appl., 26(3), 735–747, 2005
DOI: 10.1137/S0895479803436937
December 2004
https://epubs.siam.org/doi/abs/10.1137/S0895479803436937
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https://epubs.siam.org/doi/10.1137/S0895479803436937
---
https://www.researchgate.net/profile/Maher_Moakher
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https://www.researchgate.net/publication/220656766_A_Differential_Geometric_Approach_to_the_Geometric_Mean_of_Symmetric_Positive-Definite_Matrices
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This article has obtained 390 citations.
This means that it has a phenomenal success,
the subject is important,
and the article is extremely useful.
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Quotes from page 2:
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“In 1975, Anderson and Trapp [2], and Pusz and Woronowicz [20]
introduced the harmonic and geometric means
for a pair of positive operators on a Hilbert space.”
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“In [24], there was a discussion about how to define
the geometric mean of more than two
Hermitian semi-definite matrices.”
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“There have been attempts to use iterative procedures
but none seemed to work when the matrices do not commute.”
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“In [1] there is a definition for the geometric mean
of a finite set of operators,
however, the given definition is not invariant
under reordering of the matrices.”
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“The present author, while working with means
of a finite number of 3-dimensional rotation matrices [18]
discovered that there is a close connection
between the Riemannian mean of two rotations
and the geometric mean of two Hermitian definite matrices.”
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“This observation motivated the present work
on the generalization of the geometric mean
for more than two matrices using metric-based means.”
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With best regards, Jean-Claude
A11. Polynomial matrix equations
Answer 11, posted on June 6, 2019.
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Polynomials matrix equations interest
a very large number of users of matrix theory.
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This subject of research is much more difficult
than solving an ordinary polynomial equation p(x) = 0,
because matrices do not commute.
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The subject is already very difficult
for polynomials of degree 1.
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For example, given n by n matrices A, B, C, D, and E
with complex entries, find all n by n matrices X
with complex entries such that
AX + XB + CXD = E.
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One of the simplest and most well-known
such atrocity is
Sylvester equation: AX + XB = C
https://en.wikipedia.org/wiki/Sylvester_equation
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The main theorem is the following:
https://en.wikipedia.org/wiki/Sylvester_equation#Existence_and_uniqueness_of_the_solutions
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Given complex n by n matrices A, B, and C,
Sylvester equation AX + XB = C
has a unique solution X if and only if
A and -B have no common eigenvalues.
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A first problem is to extend this beautiful theorem
from matrices with entries in the field C of complex numbers
to matrices with entries in other fields.
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A first difficulties is that C has the wonderful property
to be algebraically closed, so that that all the eigenvalues
of square matrices over C are in C,
which is not the case for other fields.
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An answer to this problem was given by Jose Brox:
https://www.researchgate.net/profile/Jose_Brox
as answer 3 to the following question:
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Question posted by Yossi Peretz on May 13, 2019
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Regarding unique solution for Sylvester equation
over finite field,
is there any equivalent condition
in terms of minimal polynomials of the matrices?
https://www.researchgate.net/post/What_is_the_condition_for_unique_solution_for_the_Sylvester_equation_over_the_field_F_2
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A terrible problem is whether the theorem in answer 3
to the above question
can be extended from matrices over fields
to matrices over rings.
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With best regards, Jean-Claude
Obviously, direct methods are difficult to develop. Alternatively, to establish iterative algorithms may be more interesting., I think.
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