[Octave-bug-tracker] [bug #52533] Explanation of the rank function

[Lasse Kliemann]

URL:
  <http://savannah.gnu.org/bugs/?52533>

                 Summary: Explanation of the rank function
                 Project: GNU Octave
            Submitted by: lassekliemann
            Submitted on: Mon 27 Nov 2017 05:21:05 PM UTC
                Category: Documentation
                Severity: 3 - Normal
                Priority: 5 - Normal
              Item Group: Documentation
                  Status: None
             Assigned to: None
         Originator Name: 
        Originator Email: 
             Open/Closed: Open
         Discussion Lock: Any
                 Release: 4.2.1
        Operating System: Any

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Details:

https://www.gnu.org/software/octave/doc/interpreter/Basic-Matrix-Functions.html#Basic-Matrix-Functions

Regarding rank, it says: "The rank of a matrix is the number of linearly
independent rows or columns and determines how many particular solutions exist
to a system of equations. Use null for finding the remaining homogenous
solutions."

I find that the statement "how many particular solutions exist to a system of
equations" makes no sense. Talking about solutions only makes sense when we
have a right-hand side b also given in addition to a coefficient matrix A. It
can be said that the system Ax=b is solvable if and only if rank(A) = rank([A
b]). If it is solvable, the general solution is given by one particular
solution plus the null space. So it makes sense to mention the function 'null'
here. However, I do not see why these are the "remaining homogenous
solutions".

Here is a suggestion for the text:

"The rank of a matrix is the number of linearly independent rows or columns.
It is thus the dimension of the range space. The function 'orth' may be used
to compute an orthonormal basis of that space. For testing if a system Ax=b of
linear equations is solvable, test if rank(A) = rank([A b]). If it is
solvable, a particular solution can be found via A \ b, and the general
solution is this particular solution plus the null space, of which a basis can
be computed using null(A)."




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