See my following sentence:

So, r and lr can be regarded as obtained by a unimodular matrix
acting on the left of the original lattice basis matrix.

Is this the proper wording?

Regards,
Zhao

On 2022-09-02 08:38:50 +0000, hongy...@gmail.com said:

> See my following sentence:
>
> So, r and lr can be regarded as obtained by a unimodular matrix
> acting on the left of the original lattice basis matrix.
>
> Is this the proper wording?

What meaning are you trying to convey?

--
Athel -- French and British, living mainly in England until 1987.

On Friday, September 2, 2022 at 5:18:05 PM UTC+8, Athel Cornish-Bowden wrote:
> On 2022-09-02 08:38:50 +0000, hongy...@gmail.com said:
>
> > See my following sentence:
> >
> > So, r and lr can be regarded as obtained by a unimodular matrix
> > acting on the left of the original lattice basis matrix.
> >
> > Is this the proper wording?
> What meaning are you trying to convey?

See the following:

X is the original lattice basis matrix, which meets the following conditions:

r = u1.X
lr = u2.X
Where, both u1 and u2 are unimodular matrices, i.e., the absolute values of their determinant are always equal to 1.

All I want to express is the above.

> --
> Athel -- French and British, living mainly in England until 1987.

Best,
Zhao

On Fri, 2 Sep 2022 01:38:50 -0700 (PDT)
"hongy...@gmail.com" <hongyi.zhao@gmail.com> wrote:

> See my following sentence:
>
> So, r and lr can be regarded as obtained by a unimodular matrix
> acting on the left of the original lattice basis matrix.
>
> Is this the proper wording?
>
So, do you remember anything about context being vital when considering
usage?

You might replace 'So' with 'Therefore'. It's not an independent sentence!
If the full context was given you might even have the veracity of the
statement checked by others here.

--
Bah, and indeed Humbug.

On 02/09/22 20:13, Kerr-Mudd, John wrote:
> On Fri, 2 Sep 2022 01:38:50 -0700 (PDT) "hongy...@gmail.com"
> <hongyi.zhao@gmail.com> wrote:
>
>> See my following sentence:
>>
>> So, r and lr can be regarded as obtained by a unimodular matrix
>> acting on the left of the original lattice basis matrix.
>>
>> Is this the proper wording?
>>
> So, do you remember anything about context being vital when
> considering usage?
>
> You might replace 'So' with 'Therefore'. It's not an independent
> sentence! If the full context was given you might even have the
> veracity of the statement checked by others here.

Statements of this kind can be made much clearer by appropriate
mathematical notation. After all, mathematical notation was invented for
expressing statements that are hard to say in natural language.

For example:

"Therefore we conclude that
r = lr = UM
where U is a unimodular matrix and M is the original lattice basis matrix."

(But, lacking context, I have no idea whether this statement is correct.)

--
Peter Moylan Newcastle, NSW http://www.pmoylan.org

On Friday, September 2, 2022 at 7:12:37 PM UTC+8, Peter Moylan wrote:
> On 02/09/22 20:13, Kerr-Mudd, John wrote:
> > On Fri, 2 Sep 2022 01:38:50 -0700 (PDT) "hongy...@gmail.com"
> > <hongy...@gmail.com> wrote:
> >
> >> See my following sentence:
> >>
> >> So, r and lr can be regarded as obtained by a unimodular matrix
> >> acting on the left of the original lattice basis matrix.
> >>
> >> Is this the proper wording?
> >>
> > So, do you remember anything about context being vital when
> > considering usage?
> >
> > You might replace 'So' with 'Therefore'. It's not an independent
> > sentence! If the full context was given you might even have the
> > veracity of the statement checked by others here.
> Statements of this kind can be made much clearer by appropriate
> mathematical notation. After all, mathematical notation was invented for
> expressing statements that are hard to say in natural language.
>
> For example:
>
> "Therefore we conclude that
> r = lr = UM

The actual situation is as follows:

r = U_1.M
lr = U_2.M

In general, r != lr, and U_1 != U_2.

The context is rather mathematical, therefore, I did not involve too much here. In detail:

r = U_1.M

Here I do the Hermite normal form [1] calculation: r is the result and M is the original lattice basis matrix.

lr = U_2.M

Here I do the lattice reduction [2] calculation using the Lenstra–Lenstra–Lovász reduction algorithm: lr is the result and M is the original lattice basis matrix.
> where U is a unimodular matrix and M is the original lattice basis matrix.."
>
> (But, lacking context, I have no idea whether this statement is correct.)

[1] https://en.wikipedia.org/wiki/Hermite_normal_form
[2] https://en.wikipedia.org/wiki/Lattice_reduction
> --
> Peter Moylan Newcastle, NSW http://www.pmoylan.org

Best,
Zhao

On Friday, September 2, 2022 at 2:38:52 AM UTC-6, hongy...@gmail.com wrote:
> See my following sentence:
>
> So, r and lr can be regarded as obtained by a unimodular matrix
> acting on the left of the original lattice basis matrix.
>
> Is this the proper wording?

"Can be regarded" is pointless unless there's another way they can be
regarded. If there isn't, you could just write "are obtained".

Another problem is "/a/ unimodular matrix", which could imply that it's
the same matrix for both.

I agree with Peter Moylan that your mathematical statements are clearer
than the words. If you need the words, you might write something like
"Therefore r is obtained by multiplying the original lattice basis matrix
on the left by the unimodular matrix [whatever you call it], and lr is
obtained in the same way with the unimodular matrix [whatever you call
that one]." Or maybe this is a spot for "respectively".

I don't object to starting the sentence with "so", but some do.

--
Jerry Friedman

On Friday, September 2, 2022 at 4:38:52 AM UTC-4, hongy...@gmail.com wrote:

> See my following sentence:
>
> So, r and lr can be regarded as obtained by a unimodular matrix
> acting on the left of the original lattice basis matrix.
>
> Is this the proper wording?

Addressing just the English, and not the content, "can be
regarded as" is a sort of "hedge" -- it isn't _really_ obtained
that way, but we can pretend that / act as though it is. You
are "hedging your bet," "giving yourself an out" in case someone
complains that you didn't really obtain it that way.

On Friday, September 2, 2022 at 12:20:13 PM UTC-6, Peter T. Daniels wrote:
> On Friday, September 2, 2022 at 4:38:52 AM UTC-4, hongy...@gmail.com wrote:
>
> > See my following sentence:
> >
> > So, r and lr can be regarded as obtained by a unimodular matrix
> > acting on the left of the original lattice basis matrix.
> >
> > Is this the proper wording?
> Addressing just the English, and not the content, "can be
> regarded as" is a sort of "hedge" -- it isn't _really_ obtained
> that way, but we can pretend that / act as though it is. You
> are "hedging your bet," "giving yourself an out" in case someone
> complains that you didn't really obtain it that way.

Not in this context. From what Zhao said, it really can be obtained
that way. There might be some other way of regarding it, though.

--
Jerry Friedman

On Friday, September 2, 2022 at 3:12:44 PM UTC-4, Jerry Friedman wrote:
> On Friday, September 2, 2022 at 12:20:13 PM UTC-6, Peter T. Daniels wrote:
> > On Friday, September 2, 2022 at 4:38:52 AM UTC-4, hongy...@gmail.com wrote:
> > > See my following sentence:
> > >
> > > So, r and lr can be regarded as obtained by a unimodular matrix
> > > acting on the left of the original lattice basis matrix.
> > >
> > > Is this the proper wording?
> > Addressing just the English, and not the content, "can be
> > regarded as" is a sort of "hedge" -- it isn't _really_ obtained
> > that way, but we can pretend that / act as though it is. You
> > are "hedging your bet," "giving yourself an out" in case someone
> > complains that you didn't really obtain it that way.
>
> Not in this context. From what Zhao said, it really can be obtained
> that way. There might be some other way of regarding it, though.

Then it looks like "can be regarded as" is entirely inappropriate.

On Friday, September 2, 2022 at 11:57:49 PM UTC+8, Jerry Friedman wrote:
> On Friday, September 2, 2022 at 2:38:52 AM UTC-6, hongy...@gmail.com wrote:
> > See my following sentence:
> >
> > So, r and lr can be regarded as obtained by a unimodular matrix
> > acting on the left of the original lattice basis matrix.
> >
> > Is this the proper wording?
> "Can be regarded" is pointless unless there's another way they can be
> regarded. If there isn't, you could just write "are obtained".
>
> Another problem is "/a/ unimodular matrix", which could imply that it's
> the same matrix for both.
>
> I agree with Peter Moylan that your mathematical statements are clearer
> than the words. If you need the words, you might write something like
> "Therefore r is obtained by multiplying the original lattice basis matrix
> on the left by the unimodular matrix [whatever you call it], and lr is
> obtained in the same way with the unimodular matrix [whatever you call
> that one]." Or maybe this is a spot for "respectively".

Then, based on your above analysis, how about the following modified version?

Therefore r is obtained by multiplying the original lattice basis matrix
on the left by **one** unimodular matrix, and lr is obtained in the same way
with **another** unimodular matrix, respectively.

> I don't object to starting the sentence with "so", but some do.
>
> --
> Jerry Friedman

Best,
Zhao

On Friday, September 2, 2022 at 9:42:49 PM UTC-6, hongy...@gmail.com wrote:
> On Friday, September 2, 2022 at 11:57:49 PM UTC+8, Jerry Friedman wrote:
> > On Friday, September 2, 2022 at 2:38:52 AM UTC-6, hongy...@gmail.com wrote:
> > > See my following sentence:
> > >
> > > So, r and lr can be regarded as obtained by a unimodular matrix
> > > acting on the left of the original lattice basis matrix.
> > >
> > > Is this the proper wording?
> > "Can be regarded" is pointless unless there's another way they can be
> > regarded. If there isn't, you could just write "are obtained".
> >
> > Another problem is "/a/ unimodular matrix", which could imply that it's
> > the same matrix for both.
> >
> > I agree with Peter Moylan that your mathematical statements are clearer
> > than the words. If you need the words, you might write something like
> > "Therefore r is obtained by multiplying the original lattice basis matrix
> > on the left by the unimodular matrix [whatever you call it], and lr is
> > obtained in the same way with the unimodular matrix [whatever you call
> > that one]." Or maybe this is a spot for "respectively".

> Then, based on your above analysis, how about the following modified version?
> Therefore r is obtained by multiplying the original lattice basis matrix
> on the left by **one** unimodular matrix, and lr is obtained in the same way
> with **another** unimodular matrix, respectively.

That's good, but "respectively" should be deleted. I'd probably replace "one"
with "a" or "a certain" and "another" with "a different", but those are very fine
points.

"Respectively" means that two lists are given in corresponding order. It
could be used in a version of your sentence like

Therefore r and lr are obtained by multiplying the original lattice basis
matrix on the left by the unimodular matrices U_1 and U_2 respectively.

There "respectively" says r goes with U_1 and lr goes with U_2. It's
concise and might be a good way to say what you mean.

Constructions with "respectively" seem to appeal to many people, but in
my opinion that leads to overuse. People should use it when it's really
the best way to say something and when the correspondence isn't
obvious--not just because they like it.

--
Jerry Friedman

On Saturday, September 3, 2022 at 10:31:15 PM UTC+8, Jerry Friedman wrote:
> On Friday, September 2, 2022 at 9:42:49 PM UTC-6, hongy...@gmail.com wrote:
> > On Friday, September 2, 2022 at 11:57:49 PM UTC+8, Jerry Friedman wrote:
> > > On Friday, September 2, 2022 at 2:38:52 AM UTC-6, hongy...@gmail.com wrote:
> > > > See my following sentence:
> > > >
> > > > So, r and lr can be regarded as obtained by a unimodular matrix
> > > > acting on the left of the original lattice basis matrix.
> > > >
> > > > Is this the proper wording?
> > > "Can be regarded" is pointless unless there's another way they can be
> > > regarded. If there isn't, you could just write "are obtained".
> > >
> > > Another problem is "/a/ unimodular matrix", which could imply that it's
> > > the same matrix for both.
> > >
> > > I agree with Peter Moylan that your mathematical statements are clearer
> > > than the words. If you need the words, you might write something like
> > > "Therefore r is obtained by multiplying the original lattice basis matrix
> > > on the left by the unimodular matrix [whatever you call it], and lr is
> > > obtained in the same way with the unimodular matrix [whatever you call
> > > that one]." Or maybe this is a spot for "respectively".
>
> > Then, based on your above analysis, how about the following modified version?
> > Therefore r is obtained by multiplying the original lattice basis matrix
> > on the left by **one** unimodular matrix, and lr is obtained in the same way
> > with **another** unimodular matrix, respectively.
> That's good, but "respectively" should be deleted. I'd probably replace "one"
> with "a" or "a certain" and "another" with "a different", but those are very fine
> points.
>
> "Respectively" means that two lists are given in corresponding order. It
> could be used in a version of your sentence like
>
> Therefore r and lr are obtained by multiplying the original lattice basis
> matrix on the left by the unimodular matrices U_1 and U_2 respectively.
>
> There "respectively" says r goes with U_1 and lr goes with U_2. It's
> concise and might be a good way to say what you mean.
>
> Constructions with "respectively" seem to appeal to many people, but in
> my opinion that leads to overuse. People should use it when it's really
> the best way to say something and when the correspondence isn't
> obvious--not just because they like it.

Thank you very much for your comments and systematic analysis.
> --
> Jerry Friedman

Best,
Zhao