Linear Algebra - Science topic

If no corrections are made to the Jacobian in each iteration of Newton's method, then the search direction will become no longer the descent direction of the residuals of the equation. While this method of not updating the Jacobian exists, it does not work for my system of nonlinear equations.

I am using " Linear Algebra & Differential Equations by Stephen Goode (Author), Scott Annin (Author)"

You can have a look at the following books in this context:

1) Nonlinear Programming by Bazaara, Sherali and Shetty

2) Convex Optimization by Boyd and Vandenberghe

3) Numerical Optimization by Nocedal and Wright

4) Matrix Analysis and Applications by Xian da Zhang

Kutner, Nachtsheim, Neter, Li: Applied Linear Statistical Models

Gelman, Hill: Data Analysis Using Regression and Multilevel/Hierarchical Models

Harrell: Regression Modelling Strategies

Dobson, Barnett: Generalized Linear Models

Fox: Applied Regression Analysis & Generalized Linear Models

Yes, it is one of the best textbooks for understanding concepts of algebra mostly used for computer science as well as almost all other applied sciences.

There is no mathematical subject as (coordinator geometry), that is Coordinate Geoemetry or Analytic Geometry which is invented & designed Rene Descartes. Also sometimes it is also called Cartesian Geometry in honor of R.Descartes.

Dear Pradip Pansare , I transfer your question to GPT-4 and got such answer: "Some key trends in linear algebra research that were prevalent up to 2021 include:

I'd like to highlight topics 2, 4, 5.

Do you have a reference? I never heard of this application.

Because the ionospheric delays (and extra pseudo ranges or phases) differ at different frequencies, but the true distances are the same, by combining two equations you could exclude one variable. See "Eliminating the effect of the TEC" in Hofmann-Wellenhof et al. (2008).

Students should be shown the application of mathematics from a practical point of view. On the other hand, it is necessary to respect the theoretical side of mathematics, which is not easy for everyone to learn.@Peter Kepp Rickardo Gomes

1_S is an indicator vector

Robot manipulator design using lie algebra

Dear Krishnapriya G B

The answer is really simple.

1) Passive elements can not provide amplification. Amplification can ONLY be done with the help of active elements.

2) If we look into the I-V curve of an element, the V/I ratio must be negative in at least one of the quadrants for an active element (i.e. V/I ratio must be negative either for V>0 or V<0). This gives a non-linear curve.

NOTE: In electronics, a V-type curve is also no-linear which is treated as linear in mathematics.

Eric Welch

You can neglect the j component, so--

[0,1] dot [-2,1] = sqrt(5) * 1 * cos(theta),

so theta = 63.465 degrees counter-clockwise.

Your new unit cell is

u' = [ui * sin(63.465), uj, uk * cos(63.465)]

v' = [vi * sin(63.465), vj, vk * cos(63.465)]

w' = [wi * sin(63.465), wj, wk * cos(63.465)]

Easy... I just typed that out in about 5 minutes in the coffee shop. For more information, see Marion and Thornton, "Classical Mechanics," chapter 1 (rotation matrices are like page 4, if I remember correctly). Please correct me, if you find an error...

I school Eric every time he has a question, because he slanders me behind my back at every opportunity, with absolutely zero evidence, when I barely even know him.

The closed formula at

https://www.researchgate.net/project/Geometric-algebra-base-for-physics (under Project Log)

(A,B)_polar=A_rB_r+A_phiB_phi are attempting because the polar coordinates are orthonormal just as Cartesian. But the polar coordinates are not based on globally affine coordinates. Polar basis vectors are dependent on the position vectors. For example, unit vectors hat(r) defined at two different points (position) are different vectors (in Cartesian space R^2)-can not be overlapped by parallel transform. So if you apply for product to two different position vectors, A=A_r hat(r)+A_phi hat(phi) and B=B_r hat(r')+B_phi hat(phi'), then you will end up with (A,B)=A_r B_r hat(r)*hat(r')+etc...which its not A_rB-r+A_phiB_phi.

However, if you apply dot product to the two vectors (not position vectors) defined with respect to the same hat(r) and hat(phi) based on a common position vector, then obviously we would have (A,B)_polar=A_rB_r+A_phiB_phi.

It seems the linearization is accomplished by replacing x1, for x1^2. And separately by replacing x2, for x2^2 & x2^4.

In this way, the surface function is linearized about the origin (0,0), it means we can find f1(x1,x2)=a*x1+b*x2, whilst a and b are calculable in terms of the algebraic parameters, k and c.

But my question transforms to another level. How, we can find a compact algebraic expression for f1(x1,x2), and f2(x1,x2), close enough to the origin. This algebraic expression, need NOT be necessarily linear (it could be a nonlinear function).

1--How to find another compact analytical expression equivalent to f1(x1,x2), f2(x1,x2)? (with fair accuracy)

2-- Is it possible to find an approximation near the origin (0,0), for f1(x1,x2), f2(x1,x2), as a function of only one of the two variables (either x1, or x2)?

Regarding the second synopsis, I am to cite another ResearchGate question linked below:

However, the gist of the idea in this link is not clear to me.

In linear algebra, the rank of a matrix A is the dimension of the vector space generated (or spanned) by its columns. This corresponds to the maximal number of linearly independent columns of A. This, in turn, is identical to the dimension of the vector space spanned by its rows.

By definition, two sets are of the same cardinality if there exists a one-to-one correspondence between their elements. For a finite set, the cardinality is the number of its elements. ... For example, Z and R are infinite sets of different cardinalities while Z and Q are infinite sets of the same cardinality.

I think the question is not well posted.

B should be square and is not an additional condition!!

Also, A is a square; otherwise, there is no meaning to ask about the inverse.

In all cases, A - B is a square matrix, and det(A-B) is not zero, ensures the invertibility of A - B.

See some approaches in this question:

Till now, I do not get the book

J. M. Hoene-Wro\'nski, \emph{Introduction \`a la Philosophie des Math\'ematiques: Et Technie de l'Algorithmie}, Paris, 1811.

Dear Sir Victor Palamodov,

You are right, it is not true for the matrix. But in the question, the order of B is n*1 and order of C is 1*n. So B*C is a n*n matrix and that is a rank-one matrix.

I would recommend to ask this Q at Stack Overflow community too:

Yes, you can. The beta value and correlation value are exactly the same when the standard deviations of both variables are the same.

If we know the standard deviations of both variables and the beta value, we can easily calculate the correlation coefficient. Please check out the example below

If all righthand sides are 1, then you can reduce the problem to a series of shortest-path problems:

If not, then you can use the method in this paper:

I agree with you Norbert Tihanyi

Dear Liming, at a first glance the problem seems not to have a unique solution in general. Assuming for example that all the matrices are nxn, you need to solve n² equations (corresponding to the elements of B) in 2n² unknowns (the elements of X and Y.

If what matters is to find a solution, even if not unique, a possible approach could be to look for a minimum of the norm ||XAY-B|| with respect to the entries x_ij and y_ij. To get a differentiable function, one could take the Frobenius norm of the residual XAY-B.

The following content may be helpful

The b*b' approach, namely the LQG/LTR strategy is asymptotically singular. Thus it is intrinsically faulted.

In the context of proposal of https://www.researchgate.net/profile/Camilo_Brandao-De-Resende :

To reduce of dimensionality of space you can use Tensorized Random Projections:

https://en.wikipedia.org/wiki/Johnson%E2%80%93Lindenstrauss_lemma

A standard trick in these cases is to use the Sherman Morrison formula

However, the inverse of a sparse matrix does not have to be sparse and in particular one does not want to store inverses for sparse large matrices, so that the formula would rather have to be applied to the action A^-1b of the inverse on the right hand side of the linear system, so as to correct the solution to the original linear system A^-1b with a hopefully limited amount of operations.

Please notice that this is a very generic comment, I am sure somebody in the sparse solver community will have studied the problem in much greater depth.

You can visit the journal website and check the available options for paper submission format, if this is not clearly provided in the website you can drop an email query to the journal editor .

Have a look at https://www.youtube.com/watch?v=lAZ4tq4Ti98

https://www.flutterbys.com.au/stats/tut/tut9.2a.html

https://towardsdatascience.com/principal-component-analysis-pca-from-scratch-in-python-7f3e2a540c51

Dear Majid Zohrehbandian

Indeed it is an interesting problem. It is more profound than it seems. Your note about my counterexample is true. I have proved that your claim is valid in R³, but I am not sure about higher dimensions.

Regards

The following formally papers are related to this question:

[1] Feng Qi, Viera Cernanova, and Yuri S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, University Politehnica of Bucharest Scientific Bulletin Series A---Applied Mathematics and Physics 81 (2019), no. 1, 123--136.

[2] Feng Qi and Ai-Qi Liu, Alternative proofs of some formulas for two tridiagonal determinants, Acta Universitatis Sapientiae Mathematica 10 (2018), no. 2, 287--297; available online at https://doi.org/10.2478/ausm-2018-0022

[3] Feng Qi, Wen Wang, Dongkyu Lim, and Bai-Ni Guo, Several explicit and recurrent formulas for determinants of tridiagonal matrices via generalized continued fractions, Nonlinear Analysis: Problems, Applications and Computational Methods, Editors: Zakia Hammouch, Hemen Dutta, Said Melliani, Michael Ruzhansky; Springer Book Series Lecture Notes in Networks and Systems. The 6th International Congress of the Moroccan Society of Applied Mathematics (SM2A 2019) organized by Sultan Moulay Slimane University, Faculte des sciences et techniques, BP 523, Beni-Mellal, Morocco, during 7-9 November, 2019.

The following formally papers are related to this question:

[1] Feng Qi, Viera Cernanova, and Yuri S. Semenov, Some tridiagonal determinants related to central Delannoy numbers, the Chebyshev polynomials, and the Fibonacci polynomials, University Politehnica of Bucharest Scientific Bulletin Series A---Applied Mathematics and Physics 81 (2019), no. 1, 123--136.

[2] Feng Qi and Ai-Qi Liu, Alternative proofs of some formulas for two tridiagonal determinants, Acta Universitatis Sapientiae Mathematica 10 (2018), no. 2, 287--297; available online at https://doi.org/10.2478/ausm-2018-0022

[3] Feng Qi, Wen Wang, Dongkyu Lim, and Bai-Ni Guo, Several explicit and recurrent formulas for determinants of tridiagonal matrices via generalized continued fractions, Nonlinear Analysis: Problems, Applications and Computational Methods, Editors: Zakia Hammouch, Hemen Dutta, Said Melliani, Michael Ruzhansky; Springer Book Series Lecture Notes in Networks and Systems. The 6th International Congress of the Moroccan Society of Applied Mathematics (SM2A 2019) organized by Sultan Moulay Slimane University, Faculte des sciences et techniques, BP 523, Beni-Mellal, Morocco, during 7-9 November, 2019.

Mathematically, eigenvalues can interpret the stability of your system depending on the sign of your eigenvalues. 

I appreciate the answer of Dr. Peter Breuer .

Dear Mohammad Reza Salehizadeh

Unfortunately, the attached article

" [a] Bertsekas, Dimitri P., et al. Dynamic programming and optimal control. Vol. 1. No. 2. Belmont, MA: Athena scientific, 1995."

is full of typing errors.

General talking, we consider the linearization of the nonlinear system. Next, we study the stability of the equilibrium state of the new linear system, which indicates the nature of the stability of the nonlinear system in some neighborhoods.

The signs of the real parts of the eigenvalues of the Jacobian matrix decide which approach we should follow. We have the direct and indirect Lyapunov methods to study the stability based on the eigenvalues.

Best regards

Dear Vishwaraj B Manur,

First of all, we should separate the concept of Sampling against the concept of Sensing. These two are not interchangeable!

1. Compressed Sensing theory states that it could recover a set of coefficients (which represents in a specific transform domain the useful information from the analyzed signal) from less samples than Nyquist sampling criteria in order to be able to reconstruct a signal (of course as it could be reconstructed from uniform samples by classical Shannon theory).

2. Compressive Sampling theory states that a signal can be sampled by a protocol (non-uniform sampling, random sampling, modulation and sampling, etc.) which will allow later to be reconstructed by means of a Compressed Sensing algorithm which knows about the used sampling protocol.

3. There are at least 4 sampling ways (according to Figure 2 from https://core.ac.uk/download/pdf/34645298.pdf ) to acquire the information from a signal. Take into account that practical CS is a lossy compression, and this is due to the non-ideal process which happens when the sampling process take place.

I have another opinion though I am not certain and welcome any corrections.

I think only the Kalman gains that correspond to measurable states are between [0 1]. Those states that are not directly measurable could have Kalman gains beyond [0 1].

As your system matrix most probably is sparse, you might be better off with numerical linear algebra routines especially designed for large, sparse matrices. I did stuff on that in the late 80s, through the Cray-2 machine, but there must be more streamlined ways to do it nowadays. (In those days one had to "cut" the lengths of the vectors in order to match the smartest size of sub-vectors for the linear algebra routines, but nowadays that will most surely be automatic, to a degree, nowadays.) Talk with numerical linear algebra experts, and they will help you.

I found something more interesting here https://arxiv.org/pdf/1608.04445.pdf

The matrix is a second-order tensor. Here, all operations of tensor calculus are applicable. Acrом

I agree with Dr Ali

without sin, sinh and tanh

y = 22.1343029549693 + 10.8292764965611*x + 0.296189687976693*x*z^2 + 27*z/x^2 - 2.11016161227361*x*z - 0.270179597887278*x^2

I'm not sure of understanding the question. Nevertheless, in the case it was useful for someone, let me point put the following example: Let 0<p<q<1 and put X=l_p+L_q. The Banach envelope of X is l_1+0=l_1, which does not separate the points of X. The q-Banach envelope of X is l_q+L_q. So, X one-to-one embeds into its q-Banach envelope. In other words, the q-norm of l_q+L_q is a q-norm on X which is not equivalent to the p-norm X is equipped with.

Dear All Colleagues

The final version has been accepted for publication in the Mathematica Slovaca. For details, please click at the website:

Dear Nouran,

You can find the answer in this paper:

https://www.mdpi.com/2073-8994/12/2/243

Best regards,

Sarbast

Dear Danilo,

I just saw your answers and additional points of view. You are right concerning dimensionality. In the system is infinite dimensional because of a time delay, then we may avoid the problem by taking a discrete-time representation and using standard controllability tools. Now, as you pointed out, if the system is governed by partial differential equations the problem is different and much harder, I guess. Having said that, let me raise the following *practical* question: in the case of high-dimentional (not to mention infinte-dimensional) systems, doest it make sense, or better, is it really necessary to be able to place the system state anywhere in such a high-dimensional space? Perhaps the region of interest is a low-dimensional manifold where the controlled system should stay...

Regards.

Dear Araoluwa Filani

You need more advanced topics than linear algebra.

Different techniques in computer science are useful for doing the job. See, for example, the following article:

https://www.sciencedirect.com › topics › computer-science › nonlinear-transformation...

Best regards

Dear Liming Yuan ,

You can use the direct approach, as explained in my previous answer.

See the attached file.

Best regards

Please have a look at this attached chapter and hopefully you find your answer

Dear Parsa Ghofrani ,

According to your assumptions, we have n distinct equations.

To reach your request proceed as the following:

Rearrange the equations as follows:

1). The coefficient of the first unknown in the first equation is not zero.

2).The coefficient of the second unknown in the second equation is not zero.

etc

n).The coefficient of the nth-unknown in the nth-equation is not zero.

The new augmented matrix guaranteed your request.

Otherwise, if your matrix is full of zeros and det(A) =0, then the system has

i). No solution

or

ii) Infinite number of solutions.

Best regards

Dear Peter Breuer ,

Gershgorin disks work for columns too.

One can observe that A and A^T have the same spectrum.

Best wishes

Thanks, very useful reference

Biswajit,

In that case you are trying out some very old concepts afresh !

Not very relevant today perhaps (that's just my opinion, of course !), because control concepts themselves are a lot more mature and evolved today !!

Have you checked out total stability concepts, for instance ?

Cheers !

-Sanjay

To minimise |V-1|, where V is a positive continuous variable, just minimise x subject to x >= V-1 and x >= 1- V.

Good morning, I have studied your PDF and let me allow following comments.

1. You construct the final matrix as the sum of two matrices, these partial matrices are NOT orthogonal. Consequently, you can NOT recover these partial matrices exactly from the SVD of the final matrix, because in the SVD ALL matrices of type $\sigma_i*u_i*v_i^t $ are orthogonal!

2. There are formal errors in your PDF file - for example, the element $la(3,3)$ does not exist.

This matrix encodes all integers from 0 to 2^k-1, for given no. of rows 2^k and columns k, as binary strings.

I believe it can be named binary encoder matrix .

Dear Zeyang,

You can find in the literature elaborate comparison between the the zero forcing equalization and the MMSE. The MMSE is more accurate and suits a wide range of channel sate conditions expressed in S/N ratio. The zero forcing method is an approximate method more suitable for high signal to noise ratio where one can consider that the noise is much smaller than the received signal.

The MMSE is working well for all signal to noise ratios but it needs nuch more computational effort.

Best wishes

David E. Stewart Thanks for the tips, I appreciate your help. If you don't mind I explain a little more about the problem, I hope it comes handy. The transform I'm trying to find is called CAT (Chromatic Adaptation Transform). Typical values for such transform is as follows:

M = [0.7982 0.3389 -0.1371;

-0.5918 1.5512 0.0406;

0.0008 0.0239 0.9753]

little changes to each value above have meaningful impacts on my model. An optimized version of the matrix above could be like this:

M_Optimized = [1.2694 -0.0988 -0.1706;

-0.8364 1.8006 0.0357;

0.0297 -0.0315 1.0018]

Is there any possible tweaks to the algorithm parameter which help me to take care of little value changes? I'm afraid default particle search steps might be too large for this space. I use particleswarm command in MATLAB with default social and cognitive weights.

This transform is the main part of my model. This model tries to estimate some experimental data. Therefore my objective function is the mean error of the model in respect to experimental data. This objective function includes a non-linear function (color difference DE2000 formula) and I suppose there are multiple local minimum in the search space and these points differ slightly from each other. What could I do to make sure I'm reaching the global minimum?

I'll try the tips you mentioned. I just gave these information to clear the situation. Your further suggestions are appreciated.

One common component here is the product X'X which enters into the term to be inverted or manipulated in some way. (Called sum of squares.)

When you add a few rows, the elements of the new sum of squares matrix are the old elements plus the new sums of squares from the added rows.

See the attached small example; "yellow" = original data; "blue" = new data

There are two sides to your problem: the practical and the ambitional. You will have to look after both. Recognize the practical issues but don't let go of your ambition. You may have to get a temporary job just to live, but that does not mean you give up on your dreams.

Your problem is not unique and has been overcome by famous scientists. Faraday started working for a bookbinder and ended as a revered scientist. His personal drive got him through. Dirac got a first degree in electrical engineering and ended as a revered theorist. Einstein worked early on in a Patent office and ended as a revered theorist. Other examples can be found, such as Ramanujan. Now there's a great example of talent beating disadvantage. So you see, it's not the end of the world if there are practical difficulties in your way at this time in your life. If you keep your spirits high, focused on what really interests you, you may succeed. It may be very hard, but don't give up.

You should understand that training is not enough. You have to practice being creative. Some people on this forum will probably disagree with the following suggestion, but have a go at writing a paper on a novel topic and seeing the reaction. It may take time to find a problem that you can work on, and you may very well get rejection. But having a go will teach you more than doing a lecture course on analysis. Papers do not all have to be in quantum field theory or relativity. Go on the arXives and see what sort of topics are viable for you. Most likely, at this stage, it might be in the General Physics section. But at least you might start from there.

Good luck in your ambition. Never give up.

George Jaroszkiewicz

you can the the following book :

" Finite Dimensional Vector Spaces " , by Paul R. Halmos, in this book the co-ordinate free, axiomatic approach to linear algebra is presented.

I think my introduction threw things off track. The last part paraphrased here is the question I asked ... or ... intended to ask:

Two different methods(linear programming with a minimax objective/formulation and the Remez Exchange algorithm) can be used to achieve the same outcome. That suggests that there's a relationship between the two.

But how does the Remez Exchange algorithm (REA)and a minimax-contructed linear program (MCLP) relate in an understandable / practical mathematical context?

(I'm not able to express the relationship in this manner).

An: .......

How are they similar? How are they different?

A1: Without modification, the REA can't impose equality contraints while the MCLP can, fairly directly. Maybe that's a big deal and maybe not.

An: ......

What further extensions of the Remez algorithm might be suggested with this illumination?

Thanks dear Ryan Vogt and Debopam Ghosh

I feel you try to compute maximum eigen value and corresponding eigen vector by Power's method. Since the smallest eigen value will be near zero, it will not create any p