Timeline for Positive definite matrices diagonalised by orthogonal matrices that are also involutions
Current License: CC BY-SA 4.0
10 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 30, 2018 at 3:44 | vote | accept | GA316 | ||
| Jun 28, 2018 at 12:58 | answer | added | Rodrigo de Azevedo | timeline score: 5 | |
| Jun 28, 2018 at 11:03 | comment | added | Geoff Robinson | It's better to start with a positive definite diagonal matrix $B.$ Then for any orthogonal involution $Q,$ you see that $A = QBQ$ is still a positive definite matrix, and $QAQ = B,$ so that $A$ is a positive definite matrix diagonalized by the orthogonal involution $Q.$ Conversely, every positive definite matrix $A$ which may be diagonalized by an orthogonal involution $P$ has such a form, since $PAP$ is positive definite diagonal, and is inverted by the orthogonal involution $P.$ | |
| Jun 28, 2018 at 5:07 | comment | added | GA316 | @James But can you say anything more algebraically which would be help me more? | |
| Jun 28, 2018 at 5:06 | comment | added | GA316 | @RodrigodeAzevedo the $P$ is the question has to be symmetric as it is orthogonal and involution. That much only I can understand. If you have further insight please share with me. | |
| Jun 28, 2018 at 2:23 | comment | added | James | Geometrically, this occurs precisely when there is a reflection (through some subspace) taking the eigenspaces of A to coordinate-subspaces. | |
| Jun 28, 2018 at 0:48 | review | Close votes | |||
| Jul 2, 2018 at 3:48 | |||||
| S Jun 28, 2018 at 0:13 | history | suggested | Rodrigo de Azevedo | CC BY-SA 4.0 |
Minor improvements
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| Jun 27, 2018 at 21:21 | review | Suggested edits | |||
| S Jun 28, 2018 at 0:13 | |||||
| Jun 27, 2018 at 6:53 | history | asked | GA316 | CC BY-SA 4.0 |