http://i49.tinypic.com/34pyw50.jpg

Question is in regards to 6 (with reference to 4).

I get the hint, but am not seeing how some principal matrix being non-singular helps to prove rank of A-rI is n-1.

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- Jun 14th 2012, 08:26 PMANDS!Matrix Analysis - Horn Question
http://i49.tinypic.com/34pyw50.jpg

Question is in regards to 6 (with reference to 4).

I get the hint, but am not seeing how some principal matrix being non-singular helps to prove rank of A-rI is n-1. - Jun 15th 2012, 12:26 AMgirdavRe: Matrix Analysis - Horn Question
What you have to show is that if is a matrix which is not invertible and such that a principal submatrix of size is invertible then the rank of is . After permuting if necessary the rows and the columns, you can assume that it's the submatrix after deleting the -th row and -th column. Let the submatrix, and consider the blockwise matrix . Multipliying by , we can see that the rank of is at least . It can't be more since is not invertible.

- Jun 15th 2012, 03:13 AMANDS!Re: Matrix Analysis - Horn Question
I understand what I'm meant to show; not sure I follow your proof to be honest (syntax is a bit awkward).

Jumped on Wiki and "determinantal rank" jumped out; that seems like the easiest route with n-1 being largest possible rank of principle submatrix.