Dear Professor Feng,

Please see the attachment.

Regards,

Wiwat Wanicharpichat

Thank Wiwat a lot for the answer. Could you please give a reference in which your answer is included?

Dear Professor Wiwat Wanicharpichat

Is there a name for this matrix?

Feng Qi (F. Qi)

https://qifeng618.wordpress.com

Thank you Sir. I was solved this problem by myself with computer program. I have no any reference.

I don't know the name of this matrix.

Oh, if so, it is more nice. Can you obtain a general expression for all elements in the inverse matrix? If yes, we can jointly publish a paper. I will also do along with your idea and the above result. Perhaps, it is better that we communicate by e-mail.

Thank you Sir.

Dear Professor Feng.

Let A1, A2, ..., An denote the matrices of this form of order 1, 2, ..., n respectively. We have

A1⁻¹, A2⁻¹, ..., An-1⁻¹ are all principal submatrices of the matrix An⁻¹.

Moregenerally, Aj-1-1 is a principal submatrix of Aj-1, 2 ≤ j ≤ n.

Regards,

Wiwat Wanicharpichat

it is a binomial matrix, try the LU decomposition

Dear Hanifa, tell me more details.

I have guessed out all elements of the first row and the first column! From these elements, I can derive every elements by a recurrence relation.

I believe that I can find an explicit expression for all elements in the inverse matrix, but I need more time to write down and to prove my guess in theory.

See this link, it is not exactly your matrix, it is Pascal matrix, but it can be under this topic: binomial matrices

https://fr.wikipedia.org/wiki/Matrice_de_Pascal

it use LU decomposition. 

For LU decomposition, I can give you a short summary :

L is a Lower triangular matrix with 1 on the diagonal, U is upper

Dear Hanifa, can you compute in details?

I just see your question and know it is under that topic. I will try to compute for 3x3 and 4x4 matrix to get an idea, give me time please.

The answer of mine can be sketched by the picture uploaded with this message. When the answer is guessed out, the proof becomes easier.

Here the night is deep. See you tommorrow morning.

dear Feng, your matrix is already lower triangular, so it return to look for the inverse of A=L, while U= In.  But I don't know to compute L in general case (n any order of L). 

Also there is a link on the inverse of Pascal matrix, may be it can inspire you, with operations on binomials. 

http://math.stackexchange.com/questions/1852826/inverse-of-the-pascal-matrix

this is my file for 3x3 and 4x4 matrix, may be you will find better coefficients than mine.

Dear Wiwat

One of your recurrence relation can be reformulated as the one showed in the picture uploaded with this message.

Feng

This is a nice form.

Dear All

I guessed out all elements of the inverse matrix: all elements can be expresed by an explicit formula. See the picture below.

Its proof should not be difficult.

Feng

Dear Feng

In the explicit formula (on the picture above). What is Ci-k-1?

Wiwat Wanicharpichat

The method stated at http://math.stackexchange.com/questions/47543/getting-the-inverse-of-a-lower-upper-triangular-matrix is very useful!

Dear Wiwat, the quantity Cn stands for the Catalan number which had been mentioned above.

Thank you.

Dear All

This evening I gave a new form and a proof for the inverse of an interesting lower triangular matrix. For details, please read the PDF or PNG file uploaded.

Best regards

Feng Qi (F.Qi)

I am interested in this research, may I have to see the proof (Theorem 3.1.).

Dear Wiwat

Thank you for your interest to the research. After completing the whole paper, I will submit as a preprint somewhere. Then I will let you know the proof. Theorem 3.1 is about two identities for the Chebyshev polynomials. I believe that I can finish the whole manuscript in a week. I am now doing it speedily.

Feng

Thank you sir.

Dear All

The answer to the problem is at

http://dx.doi.org/10.20944/preprints201703.0209.v1

or

http://www.preprints.org/manuscript/201703.0209/v1

Feng

Dear All

I am preparing an "Acknowledgements" for the revised version of the following preprint. See the picture or the PDF file uploaded. Sorry for my neglect in the first version. Hope to hear your advices soon.

Feng Qi and Bai-Ni Guo, Identities of the Chebyshev polynomials, the inverse of a triangular matrix, and identities of the Catalan numbers, Preprints 2017, 22017030209, 21 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0209.v1 or at http://www.preprints.org/manuscript/201703.0209/v1.

Feng

Thank you for your kindness.

Thank you very much

Dear All

Suggest you go to https://www.researchgate.net/post/Can_one_verify_an_identity_involving_factorials to discuss the questions about coauthorship and acknowledgements.

Feng

Dear Prof. Feng Qi

I have no doctoral degree. My position is Associate Professor.

Wiwat Wanicharpichat

Dear Wiwat, 

I will remove all titles, because it will spend much time to clarify them and they are generally unnecessary in academic papers.

Feng

Up to you. Thank you sir.

DEAR SIR PROFESSOR FENG

HOPE BELOW SLIDE COULD HELP

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Dear All

It seems that the matrix and its inverse have some special and interesting properties, as described by Professor Wiwat Wanicharpichat in the eighth item among these discussions. So I suggest interesters to the matrix and its inverse to further study properties of the matrix and its inverse.

On the other hand, you can see that my proof for solving the inverse of the lower triangular matrix is indirect and not linearly algebraic. Can one find a linearly algebraic proof (a direct proof) for Theorem 4.1 in the preprint at http://dx.doi.org/10.20944/preprints201703.0209.v1? I am not good at linear algebra and matrix theory, however, you may be.

Best regards

Feng

Feng Qi, Qing Zou, and Bai-Ni Guo, Identities of the Chebyshev polynomials, the inverse of a triangular matrix, and identities of the Catalan numbers, ResearchGate Working Paper (2017), available online at http://dx.doi.org/10.13140/RG.2.2.27344.71684.

Feng Qi, Qing Zou, and Bai-Ni Guo, Some identities and a matrix inverse related to the Chebyshev polynomials of the second kind and the Catalan numbers, Preprints 2017, 2017030209, 25 pages; Available online at http://dx.doi.org/10.20944/preprints201703.0209.v2 or at http://www.preprints.org/manuscript/201703.0209/v2.

A reorganized presentation in Chinese for this preprint has been posted as

祁锋，与第二类Chebyshev多项式和Catalan数相关的一个矩阵之逆和几个恒等式， ResearchGate Presentation (2017), available online at https://doi.org/10.13140/RG.2.2.22832.46081/1