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I have a question about properties of matrices which are or are not diagonally dominant.

So I understand that a diagonally dominant Hermitian matrix with non negative diagonal entries is positive semi-definite, and that diagonally dominant implies that the matrix is non-singular.

My questions are: what can we say about a matrix where only one or two rows do not satisfy the diagonal dominance requirement; can we say anything about the data encoded in the matrix or the linear transformation the matrix represents? And is regularizing a matrix to diagonal dominance a good way to ensure invertibility?

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