Editing Diagonal matrix (section)

==Definition==

As stated above, a diagonal matrix is a matrix in which all off-diagonal entries are zero. That is, the matrix {{math|1='''D''' = (''d''<sub>''i'',''j''</sub>)}} with {{mvar|n}} columns and {{mvar|n}} rows is diagonal if
<math display="block">\forall i,j \in \{1, 2, \ldots, n\}, i \ne j \implies d_{i,j} = 0.</math>

However, the main diagonal entries are unrestricted.

The term ''diagonal matrix'' may sometimes refer to a '''{{visible anchor|rectangular diagonal matrix}}''', which is an {{mvar|m}}-by-{{mvar|n}} matrix with all the entries not of the form {{math|''d''<sub>''i'',''i''</sub>}} being zero. For example:
<math display=block>\begin{bmatrix}
  1 & 0 & 0\\
  0 & 4 & 0\\
  0 & 0 & -3\\
  0 & 0 & 0\\
\end{bmatrix} \quad \text{or} \quad \begin{bmatrix}
  1 & 0 & 0 & 0 & 0\\
  0 & 4 & 0& 0 & 0\\
  0 & 0 & -3& 0 & 0
\end{bmatrix}</math>

More often, however, ''diagonal matrix'' refers to square matrices, which can be specified explicitly as a '''{{visible anchor|square diagonal matrix}}'''. A square diagonal matrix is a [[symmetric matrix]], so this can also be called a '''{{visible anchor|symmetric diagonal matrix}}'''.

The following matrix is square diagonal matrix:
<math display="block">\begin{bmatrix}
1 & 0 & 0\\
0 & 4 & 0\\
0 & 0 & -2
\end{bmatrix}</math>

If the entries are [[real numbers]] or [[complex numbers]], then it is a [[normal matrix]] as well.

In the remainder of this article we will consider only square diagonal matrices, and refer to them simply as "diagonal matrices".