Timeline for What are known properties of matrices where off-diagonal elements are 1?
Current License: CC BY-SA 4.0
12 events
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| Dec 23, 2019 at 17:24 | comment | added | cgmil | @TerryTao Well unfortunately the formula for the determinant is recursive in nature, so I don't have a great formula for it. | |
| Dec 23, 2019 at 17:18 | comment | added | Terry Tao | You've already worked out the determinant, but in general the formula for computing the determinant of a rank one perturbation is known as the matrix determinant lemma: en.wikipedia.org/wiki/Matrix_determinant_lemma | |
| Dec 23, 2019 at 17:14 | comment | added | Terry Tao | One would have to use a limiting argument to extend the Sherman-Morrison formula to the case where the diagonal matrix in question has a single zero coefficient. As for eigenvalues, there is an identity of Lowner connecting the eigenvalues of the matrix to the entries of the diagonal matrix; see equation (24) of my recent survey on such identities with Denton, Parke, and Zhang at arxiv.org/pdf/1908.03795.pdf , as well as the equation following it. | |
| Dec 23, 2019 at 15:35 | history | became hot network question | |||
| Dec 23, 2019 at 15:27 | vote | accept | cgmil | ||
| Dec 23, 2019 at 12:49 | answer | added | Geoff Robinson | timeline score: 7 | |
| Dec 23, 2019 at 6:17 | comment | added | user6976 | @cgmil: The matrix can be invertible and the formula mat not give the inverse. For example matrix(2,1;1,1). | |
| Dec 23, 2019 at 4:47 | comment | added | cgmil | @MarkSapir That's certainly true; from the original formulation one can choose the diagonal entries to be one and then the matrix is clearly singular. If we're going to study the inverse we need to make more assumptions, but if there is an inverse it will have a certain form. (I used the Morrison formula to find that inverse.) Bonus points for other quantities such as eigenvalues/eigenvectors, though. | |
| Dec 23, 2019 at 4:44 | comment | added | user6976 | @TerryTao: This matrix may have no inverses. | |
| Dec 23, 2019 at 4:30 | comment | added | cgmil | Never seen this before, so thank you! | |
| Dec 23, 2019 at 4:12 | comment | added | Terry Tao | It's the sum of a diagonal matrix and a rank one matrix, so the Sherman-Morrison formua would give an explicit formula for the inverse. en.wikipedia.org/wiki/Sherman%E2%80%93Morrison_formula | |
| Dec 23, 2019 at 4:06 | history | asked | cgmil | CC BY-SA 4.0 |