If N is an n x n matrix and c is a (nonzero) scalar, are N and cN similar

[IliaPreston]

I have the following question please:

If N is an n x n matrix and c is a (nonzero) scalar, are N and cN similar?

Also, on a side note, I asked this question of Google (which answers these questions by AI (Artificial Intelligence)), and it gave me a response that is OBVIOUSLY wrong:
https://i.imgur.com/1vfMTQj.jpeg

Please advise.
Thanks.

[OptionBase1]

No. The eigenvalues of cN are c times (the eigenvalues of N). N and cN are similar if and only if c=1, in which case the similarity is trivial similarity since they are the exact same matrix.

[OptionBase1]

This is assuming that your use and understanding of the term "similar" when dealing with matrices is the correct one. Matrices are not similar if their eigenvalues differ.

Let A = [[1,1],[1,1]]

The eigenvalues of A = 2, 0

Let B = 2A = [[2,2],[2,2]]

The eigenvalues of B = 4, 0

Therefore, A and B, where B = cA and c = 2, are not similar.

[IliaPreston]

This is assuming that your use and understanding of the term "similar" when dealing with matrices is the correct one. Matrices are not similar if their eigenvalues differ.

Let A = [[1,1],[1,1]]

The eigenvalues of A = 2, 0

Let B = 2A = [[2,2],[2,2]]

The eigenvalues of B = 4, 0

Therefore, A and B, where B = cA and c = 2, are not similar.

Thanks for your response.
However, I am reading a linear algebra book, and one of the exercises in the book says:

If N is a nilpotent n x n matrix and c is any non-zero scalar, prove that N and cN are similar.

It is a reputable linear algebra book.
Is that exercise wrong?

Please advise.
Thanks again

[OptionBase1]

Thanks for your response.
However, I am reading a linear algebra book, and one of the exercises in the book says:

If N is a nilpotent n x n matrix and c is any non-zero scalar, prove that N and cN are similar.

It is a reputable linear algebra book.
Is that exercise wrong?

Please advise.
Thanks again

Of course, you never said in your opening post that you were specifically asking about nilpotent matrices. Your original question has been answered. Others can answer your new question if they so desire. Or you should be able to find this type of answer yourself quite easily.

Good luck.

[IliaPreston]

Of course, you never said in your opening post that you were specifically asking about nilpotent matrices. Your original question has been answered. Others can answer your new question if they so desire. Or you should be able to find this type of answer yourself quite easily.

Good luck.

Thanks for your help.

Or you should be able to find this type of answer yourself quite easily.

Unfortunately, I tried hard but couldn't find the answer.

Any help would be greatly appreciated.
Thanks again.

[OptionBase1]

The exercise is not wrong.

[OptionBase1]

And for the record, I spent 5 minutes Google searching and found relevant discussions about this question.

Sorry, not doing anyone's homework, and if this is self-study, then you might just have to skip the problem if you can't do it.

Good luck.