Editing Main diagonal (section)

==Square matrices==
For a [[square matrix]], the ''diagonal'' (or ''main diagonal'' or ''principal diagonal'') is the diagonal line of entries running from the top-left corner to the bottom-right corner.<ref>{{harvtxt|Bronson|1970|p=2}}</ref><ref>{{harvtxt|Herstein|1964|p=239}}</ref><ref>{{harvtxt|Nering|1970|p=38}}</ref> For a matrix <math> A </math> with row index specified by <math>i</math> and column index specified by <math>j</math>, these would be entries <math>A_{ij}</math> with <math>i = j</math>. For example, the [[identity matrix]] can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:
:<math>\begin{pmatrix}
 1 & 0 & 0 \\
 0 & 1 & 0 \\
 0 & 0 & 1
\end{pmatrix}</math>
The [[Trace (linear algebra)|trace of a matrix]] is the sum of the diagonal elements. 

The top-right to bottom-left diagonal is sometimes described as the ''minor'' diagonal or ''antidiagonal''.

The ''off-diagonal'' entries are those not on the main diagonal.  A ''[[diagonal matrix]]'' is one whose off-diagonal entries are all zero.<ref>{{harvtxt|Herstein|1964|p=239}}</ref><ref>{{harvtxt|Nering|1970|p=38}}</ref>

A ''superdiagonal'' entry is one that is directly above and to the right of the main diagonal.<ref>{{harvtxt|Bronson|1970|pp=203,205}}</ref><ref>{{harvtxt|Herstein|1964|p=239}}</ref> Just as diagonal entries are those <math>A_{ij}</math> with <math>j=i</math>, the superdiagonal entries are those with <math>j = i+1</math>.  For example, the non-zero entries of the following matrix all lie in the superdiagonal:
:<math>\begin{pmatrix}
 0 & 2 & 0 \\
 0 & 0 & 3 \\ 
 0 & 0 & 0
\end{pmatrix}</math>
Likewise, a ''subdiagonal'' entry is one that is directly below and to the left of the main diagonal, that is, an entry <math>A_{ij}</math> with <math>j = i - 1</math>.<ref>{{harvtxt|Cullen|1966|p=114}}</ref>  General matrix diagonals can be specified by an index <math>k</math> measured relative to the main diagonal: the main diagonal has <math>k = 0</math>; the superdiagonal has <math>k = 1</math>; the subdiagonal has <math>k = -1</math>; and in general, the <math>k</math>-diagonal consists of the entries <math>A_{ij}</math> with <math>j = i+k</math>.

A [[Band matrix|banded matrix]] is one for which its non-zero elements are restricted to a diagonal band. A [[tridiagonal matrix]] has only the main diagonal, superdiagonal, and subdiagonal entries as non-zero.