Non-zero eigenvalues and square matrix

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Discussion Overview

The discussion revolves around the properties of eigenvalues in square matrices, specifically focusing on singular and non-singular matrices. Participants explore the maximum number of non-zero eigenvalues for a singular square matrix and the implications for non-singular matrices, along with clarifications on related concepts.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Homework-related

Main Points Raised

  • One participant proposes that a singular square matrix with 7 rows can have a maximum of 6 non-zero eigenvalues, as at least one eigenvalue must be zero for the matrix to be singular.
  • Another participant agrees with the first claim and confirms that a non-singular square matrix with 5 rows can have 5 non-zero eigenvalues, since the presence of a zero eigenvalue would indicate singularity.
  • A participant seeks confirmation on their understanding that a singular square matrix cannot have an inverse due to its dimensional properties.
  • Responses indicate that the initial claims about eigenvalues are correct, but there is some confusion regarding the explanation of singular matrices.

Areas of Agreement / Disagreement

Participants generally agree on the correctness of the claims regarding the maximum number of non-zero eigenvalues for both singular and non-singular matrices. However, there is some uncertainty about the explanation of singular matrices and their properties.

Contextual Notes

Participants mention that the results discussed are elementary and commonly found in textbooks on matrix theory and linear algebra, suggesting a foundational understanding of the topic is assumed.

Who May Find This Useful

Readers interested in linear algebra, matrix theory, or those studying multivariate analysis in statistics may find this discussion relevant.

Philip Wong
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hello,

two quick question here. I've got the answer correct (I think), but I am not too sure how to explain it in words. So I hope someone tell me is my attempted explanation correct.

1) what is the maximum of non-zero eigenvalues a singular square matrix with 7 rows can have?

up to and include 6, because we wee at least 1 eigenvalues that is 0, such that it will give the determinate of the matrix will be 0. Therefore the matrix can be a singular square matrix.


2)how many non-zero eigenvalues does a non-singular square matrix with 5 rows have?
5, because if any eigenvalues in a square matrix is 0, it will turn the matrix into a singular square matrix.


by the way, I need some conformity on the idea, singular square matrix is bringing higher dimension matrix into lower dimension, such that it will not have any inverse right?
 
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Philip Wong said:
1) what is the maximum number of non-zero eigenvalues a singular square matrix with 7 rows can have?

I think you have missed the word in red :) Ya, both 1 and 2 are correct. However what you ask in the last part, I don't understand it..
 
NaturePaper said:
I think you have missed the word in red :) Ya, both 1 and 2 are correct. However what you ask in the last part, I don't understand it..

thanks! so you mean correct, as in my exaplnation was correct right? not just the answer?

don't worry about the last part of my question.
 
Yes, your explanation is correct too.

By the way, these are very much elementary results and there is no confusion that these can be found in all textbooks of matrix theory/linear algebra :)
 
well I'm doing multivariate analysis in statistics, this is part of the topics covered. so i hope i got it corrected
 

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