linear algebra: the trace of a matrix is equal to the trace of its transpose. Starting with the dataset <span class="math inline">\(\mathbf{X}\)</span>, suppose we doubly standardize it; this means we transform it so that we obtain a version <span class="math inline">\(\overset{\circ}{\mathbf{X}}\)</span> for which the columns and rows of <span class="math inline">\(\overset{\circ}{\mathbf{X}}\)</span> have zero mean and unit variance. Then, <span class="math inline">\(\overset{\circ}{\mathbf{X}}\overset{\c