Diagonal matrix

A diagonal matrix has zero entries away from its main diagonal. Its eigenvalues are its diagonal entries.

For a diagonal matrix $D=\mathrm{diag}(d_1,\dots ,d_n)$ over a field, matrix multiplication gives $(BD-DB{)}_{ij}=(d_j-d_i)b_{ij}$. Hence a matrix $B$ commutes with $D$ exactly when $b_{ij}=0$ wherever $d_i\ne d_j$. If all diagonal entries are distinct, the centralizer of a diagonal matrix consists exactly of diagonal matrices. When entries repeat, the allowed entries form unrestricted blocks on the corresponding eigenspaces.

A scalar matrix is a scalar multiple of the identity matrix, equivalently a diagonal matrix with all diagonal entries equal. It commutes with every square matrix of the same size. In $G{L}_n(F)$, the nonzero scalar matrices form the central subgroup removed to define the projective linear group.