main diagonal

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.{{harvtxt|Bronson|1970|p=2}}{{harvtxt|Herstein|1964|p=239}}{{harvtxt|Nering|1970|p=38}} For a matrix $A$ with row index specified by $i$ and column index specified by $j$, these would be entries $A\_\{ij\}$ with $i\; =\; j$. For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:

:$\backslash begin\{pmatrix\}$

1 & 0 & 0 \\

0 & 1 & 0 \\

0 & 0 & 1

\end{pmatrix}

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.{{harvtxt|Herstein|1964|p=239}}{{harvtxt|Nering|1970|p=38}}

A superdiagonal entry is one that is directly above and to the right of the main diagonal.{{harvtxt|Bronson|1970|pp=203,205}}{{harvtxt|Herstein|1964|p=239}} Just as diagonal entries are those $A\_\{ij\}$ with $j=i$, the superdiagonal entries are those with $j\; =\; i+1$. For example, the non-zero entries of the following matrix all lie in the superdiagonal:

:$\backslash begin\{pmatrix\}$

0 & 2 & 0 \\

0 & 0 & 3 \\

0 & 0 & 0

\end{pmatrix}

Likewise, a subdiagonal entry is one that is directly below and to the left of the main diagonal, that is, an entry $A\_\{ij\}$ with $j\; =\; i\; -\; 1$.{{harvtxt|Cullen|1966|p=114}} General matrix diagonals can be specified by an index $k$ measured relative to the main diagonal: the main diagonal has $k\; =\; 0$; the superdiagonal has $k\; =\; 1$; the subdiagonal has $k\; =\; -1$; and in general, the $k$-diagonal consists of the entries $A\_\{ij\}$ with $j\; =\; i+k$.

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.

{{see also|Anti-diagonal matrix}}

The antidiagonal (sometimes counter diagonal, secondary diagonal (*), trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order $N$ square matrix $B$ is the collection of entries $b\_\{i,j\}$ such that $i\; +\; j\; =\; N+1$ for all $1\; \backslash leq\; i,\; j\; \backslash leq\; N$. That is, it runs from the top right corner to the bottom left corner.

:$\backslash begin\{bmatrix\}$

0 & 0 & \color{red}{1}\\

0 & \color{red}{1} & 0\\

\color{red}{1} & 0 & 0\end{bmatrix}

(*) 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., $A\_\{i,\backslash ,i\backslash pm\; k\}$ 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.

• Trace

{{reflist}}

• {{ 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)

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