Centered PSD matrices with thin spectrum are M-matrices
We show that real, symmetric, centered (zero row sum) positive semidefinite matrices of order n and rank n − 1 with eigenvalue ratio λ max / λ min ≤ n / ( n − 2 ) between the largest and smallest nonzero eigenvalue have nonpositive off-diagonal entries, and that this eigenvalue criterion is ti...
| Main Author: | |
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| Format: | Journal article |
| Language: | English |
| Published: |
University of Wyoming Libraries
2023
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| Summary: | We show that real, symmetric, centered (zero row sum) positive semidefinite matrices of order
n
and rank
n
−
1
with eigenvalue ratio
λ
max
/
λ
min
≤
n
/
(
n
−
2
)
between the largest and smallest nonzero eigenvalue have nonpositive off-diagonal entries, and that this eigenvalue criterion is tight. The result is relevant in the context of matrix theory and inverse eigenvalue problems, and we discuss an application to Laplacian matrices. |
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