Diagonal matrix

In linear algebra, a diagonal matrix is a matrix whose only non-zero entries are those on the main diagonal (top left to bottom right). Thus, the matrix D = (d_i,j) is diagonal if:

$d_{i,j}=0{\mbox{ if }}i\neq j$

Example:

${\begin{bmatrix}1&0\\0&4\end{bmatrix}}$

Any diagonal matrix is also a symmetric matrix. The identity matrix I_n is diagonal.

Diagonal matrices occur in many areas of linear algebra. Matrix multiplication of diagonal matrices is very simple, so if a matrix we are interested in can in some way be replaced by a diagonal one, computations involving it are much faster.

If D, E are diagonal, and their product DE = F = (f_i,j) then:

$f_{i,j}=d_{i,j}e_{i,j}$

in other words, diagonal matrices can simply be multiplied entry by entry, because the zeros cancel all the other parts of the multiplication formula.

See also diagonalization.