%%%%%%%%%%%%%%%%%%%%%%%%%

% John P Cunningham

% 2010

%

% reshapeSkew.m

%

% jPCA2 support function

%

% A skew-symmetric matrix xMat of size n by n really only

% has n*(n-1)/2 unique entries.  That is, the diagonal is 0, and the

% upper/lower triangle is the negative transpose of the lower/upper.  So,

% we can just think of such a matrix as a vector x of size n(n-1)/2.  This

% function reshapes such a vector into the appropriate skew-symmetric

% matrix.  

% 

% The required ordering in x is row-minor, namely that xMat(1,1) = 0,

% xMat(2,1) = x(1), xMat(3,1) = x(2), and so on.

%

% This function goes either from vector to matrix or vice versa, depending

% on what the argument x is.

%

% In short, this function just reindexes a vector to a matrix or vice

% versa.

%%%%%%%%%%%%%%%%%%%%%%%%

function [ Z ] = reshapeSkew( x )

    % this reshapes a n(n-1)/2 vector to a n by n skew symmetric matrix, or vice versa.

    % First we must check if x is a matrix or a vector.

    if isvector(x)

        % then we are making a matrix

        % first get the size of the appropriate matrix

        % this should be n(n-1)/2 entries.

        % this is the positive root

        n = (1 + sqrt(1 + 8*length(x)))/2;

        % error check

        if n~=round(n) % if not an integer

            % this is a bad argument

            fprintf('ERROR... the size of the x vector prevents it from being shaped into a skew symmetric matrix.\n');

            keyboard;

        end

        % now make the matrix

        % initialize the return matrix

        Z = zeros(n);

        % and the marker index

        indMark = 1;

        for i = 1 : n-1

            % add the elements as appropriate.

            Z(i+1:end,i) = x(indMark:indMark+(n-i)-1);

            % now update the index Marker

            indMark = indMark + (n-i);

        end

        % now add the skew symmetric part

        Z = Z - Z';

    else

        % then we are making a vector from a matrix (note that the 

        % standard convention of lower case being a vector and upper case

        % being a matrix is now reversed).

        % first check that everything is appropriately sized and skew

        % symmetric

        if size(x) ~= size(x')

            % this is not symmetric

            fprintf('ERROR... the matrix x is not square, let alone skew-symmetric.\n');

            keyboard;

        end

        % now check for skew symmetry

        if abs(norm(x + x'))>1e-8

            % this is not skew symmetric.

            fprintf('ERROR... the matrix x is not skew-symmetric.\n');

            keyboard;

        end

        % everything is ok, so take the size

        n = size(x,1);

        % now make the vector Z

        indMark = 1;

        for i = 1 : n-1

            % add the elements into a column vector as appropriate.

            Z(indMark:indMark+(n-i)-1,1) = x(i+1:end,i);

            % now update the index Marker

            indMark = indMark + (n-i);

        end

    end

end