Editing Main diagonal

{{short description|Entries of a matrix for which the row and column indices are equal}}

In [[linear algebra]], the '''main diagonal''' (sometimes '''principal diagonal''', '''primary diagonal''', '''leading diagonal''', '''major diagonal''', or '''good diagonal''') of a [[matrix (mathematics)|matrix]] <math>A</math> is the list of entries <math>a_{i,j}</math> where <math>i = j</math>. All [[off-diagonal element]]s are [[zero]] in a [[diagonal matrix]]. The following four matrices have their main diagonals indicated by red ones: 

<math do not display=block>\begin{bmatrix}
\color{red}{1} & 0 & 0\\
0 & \color{red}{1} & 0\\
0 & 0 & \color{red}{1}\end{bmatrix}
\qquad
\begin{bmatrix}
\color{red}{1} & 0 & 0 & 0 \\
0 & \color{red}{1} & 0 & 0 \\
0 & 0 & \color{red}{1} & 0 \end{bmatrix}
\qquad
\begin{bmatrix}
\color{red}{1} & 0 & 0 \\
0 & \color{red}{1} & 0 \\
0 & 0 & \color{red}{1} \\
0 & 0 & 0
\end{bmatrix}

\qquad
\begin{bmatrix}
\color{red}{1} & 0 & 0 & 0 \\
0 & \color{red}{1} & 0 & 0 \\
0 & 0 & \color{red}{1} & 0 \\
0 & 0 & 0 & \color{red}{1} 
\end{bmatrix}
 </math>

==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.

==Antidiagonal==

{{see also|Anti-diagonal matrix}}

The '''antidiagonal''' (sometimes '''counter diagonal''', '''secondary diagonal''' (*), '''trailing diagonal''', '''minor diagonal''', '''off diagonal''', or '''bad diagonal''') of an order <math>N</math> [[square matrix]] <math>B</math> is the collection of entries <math>b_{i,j}</math> such that <math>i + j = N+1</math> for all <math>1 \leq i, j \leq N</math>. That is, it runs from the top right corner to the bottom left corner.
:<math>\begin{bmatrix}
0 & 0 & \color{red}{1}\\
0 & \color{red}{1} & 0\\
\color{red}{1} & 0 & 0\end{bmatrix}</math>

(*) '''''Secondary''''' (as well as ''trailing'', ''minor'' and ''off'') diagonals very often also mean the (a.k.a. ''k''-th) diagonals '''''parallel''''' to the main or principal diagonals, ''i.e.'', <math>A_{i,\,i\pm k}</math> for some nonzero k =1, 2, 3, ... More generally and universally, the '''''off diagonal''''' elements of a matrix are all elements ''not'' on the main diagonal, ''i.e.'', with distinct indices ''i &ne; j''.

== See also ==
* [[Trace (matrix)|Trace]]

== Notes ==

{{reflist}}

== References ==
* {{ citation | first1 = Richard | last1 = Bronson | year = 1970 | lccn = 70097490 | title = Matrix Methods: An Introduction | publisher = [[Academic Press]] | location = New York }}
* {{ citation | first1 = Charles G. | last1 = Cullen | title = Matrices and Linear Transformations | location = Reading | publisher = [[Addison-Wesley]] | year = 1966 | lccn = 66021267 }}
* {{ citation | first1 = I. N. | last1 = Herstein | year = 1964 | isbn = 978-1114541016 | title = Topics In Algebra | publisher = [[Blaisdell Publishing Company]] | location = Waltham }}
* {{ citation | first1 = Evar D. | last1 = Nering | year = 1970 | title = Linear Algebra and Matrix Theory | edition = 2nd | publisher = [[John Wiley & Sons|Wiley]] | location = New York | lccn = 76091646 }}
* {{MathWorld|id=Diagonal|title=Main diagonal}}

[[Category:Matrices (mathematics)]]


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