Just to spell it out, am I right in thinking that you can scale all the matrix elements so that each entry on the diagonal is the sign of its original value, i.e. −1 or 1, but if the resulting diagonal contains any −1s then it can’t be PSD?
Yes indeed. One definition of a PSD matrix is some matrix M such that, for any vector X, XTMX≥0 (so, it defines some kind of scalar product).
If Mi,i=y, then you can always divide the whole row/column by √y, which is equivalent to applying some scaling, this won’t change the fact that M is PSD or not.
If Mi,i=−1, then if you try the vector X:∀j≠i,Xj=0;Xi=1, you can check that XTMX=−1, thus the matrix isn’t PSD.
Just to spell it out, am I right in thinking that you can scale all the matrix elements so that each entry on the diagonal is the sign of its original value, i.e. −1 or 1, but if the resulting diagonal contains any −1s then it can’t be PSD?
Yes indeed. One definition of a PSD matrix is some matrix M such that, for any vector X, XTMX≥0 (so, it defines some kind of scalar product).
If Mi,i=y, then you can always divide the whole row/column by √y, which is equivalent to applying some scaling, this won’t change the fact that M is PSD or not.
If Mi,i=−1, then if you try the vector X:∀j≠i,Xj=0;Xi=1, you can check that XTMX=−1, thus the matrix isn’t PSD.