How I can find in a big matrix the elements that are simultaneously min on rows and max on columns?

For example I have

A=rand(50,50);

And I want to find the position and the values that are simultaneously min on rows and max on columns

[Azzi Abdelmalek]

What if you have two minimum in the same row?

[Image Analyst]

How can that happen, unless he would have integers? Each row will have exactly one element that is the min. Same for the columns. Of course being the min element on a row does not mean that same element is also the min element of the column it is in - in fact, it probably won't be.

[Azzi Abdelmalek]

He said for example A=rand(50). It's just an example

[Image Analyst]

Exactly. So there will never be two minimums in the same row. And, for a given row, whatever column the row min does occur in will only be the max of the column it's in 1 time in 50. So most of the time he won't find any elements meeting the criteria. Perhaps you and I are interpreting what he said differently.

Answer 1

[Image Analyst]

This is very unlikely for a 5 by 5. I decreased the matrix to a 4 by 3 and it happens sometimes. Here's the code:

A = randn(4, 3)
[rowMins,colIndexOfMins] = min(A,[],2) % Within rows
[colMaxs, rowIndexOfMaxs] = max(A,[],1) % Within columns
rowMinMap = ismember(A, rowMins)
colMaxMap = ismember(A, colMaxs)
itsBoth = rowMinMap & colMaxMap
% Tell if there are any
if any(itsBoth(:))
  [rows, cols] = find(itsBoth);
  for k = 1 : length(rows)
    message = sprintf('Row %d, column %d is both a row min and a column max',...
      rows(k), cols(k));
      helpdlg(message);
  end
else
  message = 'No element is both a row min and a column max';
  warndlg(message);
end

In the command window for one case where it found such an element:

A =
        -0.636128249660041         0.777003526719788          1.04858076053644
         0.317851419059697         0.622393924172013         0.660707086367175
         0.138047974449526         0.647380884516047          2.50877247318511
        -0.710735074811226        -0.425631681660351          1.06345963904102
rowMins =
        -0.636128249660041
         0.317851419059697
         0.138047974449526
        -0.710735074811226
colIndexOfMins =
     1
     1
     1
     1
colMaxs =
         0.317851419059697         0.777003526719788          2.50877247318511
rowIndexOfMaxs =
     2     1     3
rowMinMap =
     1     0     0
     1     0     0
     1     0     0
     1     0     0
colMaxMap =
     0     1     0
     1     0     0
     0     0     1
     0     0     0
itsBoth =
     0     0     0
     1     0     0
     0     0     0
     0     0     0

Answer 2

[Matt J]

 rowmin=min(A,[],2);
 colmax=max(A,[],1);

 locations = bsxfun(@eq,A,rowmin) & bsxfun(@eq,A,colmax);