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 <math>A</math> is the list of entries <math>a_{i,j}</math> where <math>i = j</math>. All off-diagonal elements 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 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 '''{{visible anchor|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 '''{{visible anchor|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 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==

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 ≠ j''.

== See also == * Trace

== Notes ==

== 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 = Wiley | location = New York | lccn = 76091646 }} * {{MathWorld|id=Diagonal|title=Main diagonal}}

Category:Matrices (mathematics)