Talk:Inverse element

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 Field: Algebra

left/right matrix inverse[edit]

I think the phrase

If the determinant of M is zero, it is impossible for it to have a one-sided inverse; therefore a left inverse or right inverse implies the existence of the other one.

is redundant, and not very clear, (IMHO):

  1. why det=0 => no one-sided inverse ?
  2. why "no one-sided inverse" => "l/r inverse implies existence of the other" ?
  3. more specifically, why is a left inverse matrix also is a right inverse ?

But as I already did some "cutting", I'll leave it for the moment... MFH 15:17, 5 Apr 2005 (UTC)


I disagree...

This problem stated on this page is slightly confusing to me:


  \begin{bmatrix}
    17 & 22 & 27 \\
    22 & 29 & 36\\
    27 & 36 & 45
  \end{bmatrix}

The statement above that this "Is a singular matrix, and can't be inverted." is partly incorrect. An inverse is shown here:


  \begin{bmatrix}
    1.25 & 0.0 & -0.75 \\
    0.0 & 0.0 & 0.0 \\
    -0.75 & 0.0 & 0.47222
  \end{bmatrix}

This matrix is a left and right hand inverse (Moore-Penrose properties 1 and 2).

The null-space for this matrix is:


  \begin{bmatrix}
    0.0 & -0.5 & 0.0 \\
    0.0 & 1.0 & 0.0 \\
    0.0 & -0.5 & 0.0
  \end{bmatrix}

There are an infinite number of solutions to the underspecified or singular matrix. The calculations are not difficult and are described on the http://mjollnir.com/matrix/demo.html page. Should I create a wikipedia page to describe the algorithm?

The original matrix was:


  \begin{bmatrix}
    1 & 2 & 3 \\
    4 & 5 & 6
  \end{bmatrix}

Using the http://mjollnir.com/matrix/demo.html site online calculator, a pseudo-inverse is directly calculated:


  \begin{bmatrix}
    -1.0 & 0.5 \\
    0.0 & 0.0 \\
    0.666667 & -0.166667
  \end{bmatrix}

The null space is the same as the above:


  \begin{bmatrix}
    0.0 & -0.5 & 0.0 \\
    0.0 & 1.0 & 0.0 \\
    0.0 & -0.5 & 0.0
  \end{bmatrix}

rand huso (talk) 18:08, 23 February 2008 (UTC)