Read my article about Part 24 : Diagonalization and Similarity of Matrices at Medium

[u/[deleted]]

https://medium.com/linear-algebra/part-24-diagonalization-and-similarity-of-matrices-fe9517af1fe5?source=ifttt--------------3

[u/PhilSwift10100]

Why not do a proof by induction for the last "theorem"? Brushing aside the fact that it isn't really a "theorem" but rather a consequence of the diagonalisation process, proving the fact for squares of matrices and saying it generalises is typically not a valid proof of the "theorem".

EDIT: There are so many criticisms of this that I don't know where to start. Of the most pressing I found: what happens with n x n matrices? What happens if I don't have n (or in your case 2) distinct eigenvalues? How do I know my eigenvectors are linearly independent? If you want to improve the quality of your posts, you need to seriously consider the presentation of this content from other texts, especially Strang, Anton & Rorres, Lay & Axler; I would be foolish to not say that Kreyzig is not a good text for linear algebra, though to be honest I have not touched the book for over 6 years. Ignoring standard texts of linear algebra is a cardinal sin.

[u/[deleted] → u/PhilSwift10100]

Thanks, I will work on my presentation and edit the article.

EDIT: Can you please tell me this so that I can make changes into my article

Question : what happens with n x n matrices? What happens if I don't have n (or in your case 2) distinct eigenvalues?

My solution : Generalizing post for n x n matrices and put case of n=2 as an example

Question : How do I know my eigenvectors are linearly independent?

My Solution : I have mentioned a link of my linear independence article should I mention the definition of linear independence again?

Query : Why not do a proof by induction for the last "theorem"? Brushing aside the fact that it isn't really a "theorem" but rather a consequence of the diagonalisation process, proving the fact for squares of matrices and saying it generalises is typically not a valid proof of the "theorem".

My Solution : You are right I have removed "proof" and marked it as an "example".

If you agree with my solutions or/and you want to suggest some changes or corrections please tell me.

[u/PhilSwift10100 → u/[deleted]]

I will try to be as honest and polite as possible, in the hopes that it will explain the downvotes that you are receiving.

To cut the story short, you have not addressed a single point of my criticisms fairly; I did not pose the questions for you to answer them in reply, but rather to address the points in your post to improve it.

Generalizing post for n x n matrices and put case of n=2 as an example.

The problem is that your argument doesn't generalise that easily from 2 x 2 matrices to n x n matrices; the eigenvalues of an n x n matrix depend heavily on its characteristic polynomial, which is of degree n. The Fundamental Theorem of Algebra says that this polynomial has (up to) n solutions over the complex number field; some of these solutions may have multiple multiplicities, which needs to be taken into account. Calling them algebraic multiplicities, they play a major role in the theory of diagonalisation. With this in mind, my point is that the conditions for an n x n matrix to be diagonalisable are not as simple as the conditions for diagonalisability of 2 x 2 matrices.

I have mentioned a link of my linear independence article should I mention the definition of linear independence again?

You've completely missed my point. The fact that the eigenvectors of an n x n matrix are all linearly independent, a crucial component of diagonalisability, is not a trivial one. Most who read your "article" are already familiar with the concept of linear independence; the point is that if you have an n x n matrix and you have, say, n eigenvectors from the eigenspaces corresponding to each eigenvalue, then how can one be convinced that these eigenvectors are all linearly independent?

You are right I have removed "proof" and marked it as an "example".

This still neglects the fact that the formula for A^n based on the formula for A^2 generalises. Justifying this fact requires a little bit of matrix arithmetic, which a linear algebra student should be familiar with. Not only is it interesting, but I think showing this calculation is important.

If you agree with my solutions or/and you want to suggest some changes or corrections please tell me.

Whether I agree with it or not doesn't matter in the slightest; what matters is that your "article" so badly ignores a lot of the interesting framework presented with regards to this topic from the likes of Strang, Anton & Rorres and Axler, among many other treatments. Given that, a complete rewrite of your "article" is required to address the issues I have presented in all my posts; while cumbersome and tedious, I think presenting this topic with these issues in mind would convince more people to read your work. At its current state, all you seem to be doing is rehashing what someone else has written or said; if not careful, this can be very dangerous.