Timeline for The "matrix direct sum" monoid modulo unitary equivalence
Current License: CC BY-SA 4.0
15 events
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| Nov 10, 2021 at 19:25 | history | edited | wlad | CC BY-SA 4.0 |
clarified that the matrices can have any shape
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| Nov 10, 2021 at 19:16 | comment | added | wlad | @BenjaminSteinberg The direct sum $M \oplus 0_{1 \times 0}$ pads the matrix $M$ with an additional row of zeroes. Similarly, the direct sum $M \oplus 0_{0 \times 1}$ pads $M$ with an additional column of zeroes. | |
| Nov 10, 2021 at 19:12 | comment | added | wlad | @BenjaminSteinberg Yes. Non-square matrices are allowed | |
| Nov 10, 2021 at 19:12 | comment | added | wlad | @BenjaminSteinberg A $0 \times n$ matrix represents a linear map from $R^0$ to $R^n$. There is only one such map. For each $n$, the map is unique, but for different $n$ they are different. | |
| Nov 10, 2021 at 19:10 | comment | added | Benjamin Steinberg | How is a 0xn matrix different than a 0x0. Am I write that M(R,*) allows nonsquare matrices? | |
| Nov 10, 2021 at 19:07 | history | edited | wlad | CC BY-SA 4.0 |
added 2 characters in body
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| Nov 10, 2021 at 18:58 | history | edited | wlad | CC BY-SA 4.0 |
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| Nov 10, 2021 at 13:34 | history | edited | wlad | CC BY-SA 4.0 |
edited title
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| Nov 10, 2021 at 10:47 | comment | added | wlad | The $0 \times 1$ and $1 \times 0$ generators allow for non-square and non-invertible matrices to have SVDs. The multiset of singular values can include those 2 matrices as well. The matrix $(0)$ on the other hand needs to be excluded from the generators. | |
| Nov 10, 2021 at 10:43 | comment | added | wlad | @BenjaminSteinberg Probably by me, especially if it was asked recently. I got it wrong a few times. In my first formulation, I didn't consider any matrices with zero rows or zero columns except for the (obviously unique) $0 \times 0$ matrix. I realise now that I need to consider $0 \times n$ and $n \times 0$ matrices as well. In the case where $(R,*) = (\mathbb C, a + bi \mapsto a - bi)$, we have that the generators of $M(R,*)$ are the $1 \times 1$ matrices with positive entries as well as the unique $0 \times 1$ and $1 \times 0$ matrices. | |
| Nov 10, 2021 at 10:40 | comment | added | Benjamin Steinberg | I’m pretty sure I remember this question being asked in the past but I couldn't find it searching | |
| Nov 10, 2021 at 10:20 | history | edited | wlad | CC BY-SA 4.0 |
added 160 characters in body
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| Nov 10, 2021 at 10:13 | history | edited | wlad | CC BY-SA 4.0 |
forgot to quotient by $\sim$
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| Nov 10, 2021 at 9:21 | history | edited | wlad | CC BY-SA 4.0 |
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| Nov 10, 2021 at 9:15 | history | asked | wlad | CC BY-SA 4.0 |