Discussion Overview
The discussion revolves around the relationship between eigenvalues and singular values of a matrix, specifically under the condition that the matrix is symmetric positive definite. Participants are exploring the proof of this relationship and the implications of symmetry and positive definiteness.
Discussion Character
- Technical explanation
- Debate/contested
- Mathematical reasoning
Main Points Raised
- One participant seeks to prove that a matrix A has eigenvalues equal to its singular values if and only if it is symmetric positive definite, expressing uncertainty about the symmetry aspect.
- Another participant suggests that posting existing work would be beneficial for further discussion.
- A participant notes that if a singular value is an eigenvalue of \( A^TA \), then A must be positive semi-definite, questioning the starting point regarding symmetry.
- One participant references the Spectral Theorem, stating that every real symmetric matrix is diagonalizable and thus has real eigenvalues and corresponding eigenvectors.
- Another participant challenges the problem statement, suggesting that the assumption of positive definiteness typically includes symmetry and points out that singular values can be zero, which complicates the claim about eigenvalues being positive.
- This participant agrees that if A is symmetric positive definite, then its eigenvalues are positive and relates the eigenvalues of A to the singular values through the relationship with \( A^TA \).
Areas of Agreement / Disagreement
Participants express differing views on the necessity of symmetry in the context of positive definiteness and the implications of singular values being zero. There is no consensus on the correctness of the initial claim regarding eigenvalues and singular values.
Contextual Notes
Participants have not fully resolved the assumptions regarding the definitions of positive definiteness and symmetry, nor have they clarified the implications of singular values being zero on the eigenvalues of A.