Re: Rank of a matrix depending on a variable

• To: mathgroup at smc.vnet.net
• Subject: [mg122017] Re: Rank of a matrix depending on a variable
• From: Ray Koopman <koopman at sfu.ca>
• Date: Sun, 9 Oct 2011 03:52:13 -0400 (EDT)
• Delivered-to: l-mathgroup@mail-archive0.wolfram.com
• References: <j6q5ue\$p0o\$1@smc.vnet.net>

```On Oct 8, 11:48 am, Mikel <ketakop... at gmail.com> wrote:
> Hi,
>
> A bit of background for my problem: I'm trying to solve a linear
> system of equations in matrix form, A.x==B. I have the A matrix,
> which is square and singular. I also have the B vector, which
> depends on some parameters (or variables, I'm not sure how to call
> it). I'm looking for the value of the parameters, so there exists
> solution to the system. In other words, I want the rank of the
> augmented matrix AB to be equal to the rank of A.
>
> I thought about using the MatrixRank function, but it does not
> support parameters, it seems. Then I tried RowReduce, but even
> though in the first example of the help page the result is given
> with parameters, I only get numbers.
>
> Here's an example to try:
>
> ============
>
> Clear[a]
> A = {{1, 0}, {2, 0}};
> B = {{a}, {2}};
> AB = ArrayFlatten[{{A, B}}];
> MatrixRank[A]
> MatrixRank[AB]
> RowReduce[AB]
>
> a = 1;
> MatrixRank[A]
> MatrixRank[AB]
> RowReduce[AB]
>
> ============
>
> As can be seen, the rank does depend on the value of the
> "a" parameter, but RowReduce just outputs a numeric value.
>
> So, the questions are:
> 1) Why does RowReduce not give a parameter in the result?
>    Is this a bug?
> 2) Is there any other way to solve the problem, i.e.
>    to get the rank of AB as a function of the "a" variable?
>
> I'm using Mathematica 7, Student version, for what is worth.

B must be orthogonal to the nullspace of the columns of A.

In:= A = {{1,0},{2,0}};
B = {{a},{2}};
Solve[NullSpace@Transpose@A.B == {{0}}, a]

Out= {{a -> 1}}

```

• Prev by Date: RE: Sort on vector component
• Next by Date: Re: "Esoteric of the week"
• Previous by thread: Rank of a matrix depending on a variable
• Next by thread: RE: Rank of a matrix depending on a variable
{"ft_lang_label":"__label__en","ft_lang_prob":0.808789,"math_prob":0.9075231,"size":1908,"snap":"2020-24-2020-29","text_gpt3_token_len":590,"char_repetition_ratio":0.121323526,"word_repetition_ratio":0.05602241,"special_character_ratio":0.3501048,"punctuation_ratio":0.1904762,"nsfw_num_words":0,"has_unicode_error":false,"math_prob_llama3":0.98838836,"pos_list":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14],"im_url_duplicate_count":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"WARC_HEADER":"{\"WARC-Type\":\"response\",\"WARC-Date\":\"2020-07-03T15:12:54Z\",\"WARC-Record-ID\":\"<urn:uuid:fc348add-a162-4797-85f7-2bfb9a213111>\",\"Content-Length\":\"45429\",\"Content-Type\":\"application/http; msgtype=response\",\"WARC-Warcinfo-ID\":\"<urn:uuid:f631b380-1594-4308-b437-d6a86e193d31>\",\"WARC-Concurrent-To\":\"<urn:uuid:88d718f4-5157-4467-b8fa-5ffd3832f942>\",\"WARC-IP-Address\":\"140.177.205.73\",\"WARC-Target-URI\":\"http://forums.wolfram.com/mathgroup/archive/2011/Oct/msg00253.html\",\"WARC-Payload-Digest\":\"sha1:UFA2SYEI6KA5P6PCTOJH7AK3N63U5RHB\",\"WARC-Block-Digest\":\"sha1:ONZ2U26SQ7AQH5MR6O4HJJNDLE5TMR76\",\"WARC-Identified-Payload-Type\":\"text/html\",\"warc_filename\":\"/cc_download/warc_2020/CC-MAIN-2020-29/CC-MAIN-2020-29_segments_1593655882051.19_warc_CC-MAIN-20200703122347-20200703152347-00514.warc.gz\"}"}