diagonal matrix

A square matrix whose only nonzero entries (if any) lie on the main diagonal.

A diagonal matrix $D=\mathrm{diag}(d_1,\dots,d_n)$ is any square matrix whose only nonzero entries, if any, lie on the main diagonal. Every identity matrix is diagonal (with every $d_i=1$ ), but a diagonal matrix's entries can be anything, including zeros. A matrix that is both upper and lower triangular is exactly a diagonal matrix.