main diagonal

In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix $A$ is the list of entries $a_$ where $i = j$. All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones:

$\begin \color & 0 & 0\\ 0 & \color & 0\\ 0 & 0 & \color\end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 & \color & 0 \end \qquad \begin \color & 0 & 0 \\ 0 & \color & 0 \\ 0 & 0 & \color \\ 0 & 0 & 0 \end \qquad \begin \color & 0 & 0 & 0 \\ 0 & \color & 0 & 0 \\ 0 & 0 & \color & 0 \\ 0 & 0 & 0 & \color \end$

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. For a matrix $A$ with row index specified by $i$ and column index specified by $j$, these would be entries $A_$ with $i = j$. For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:
:$\begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end$
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.

A entry is one that is directly above and to the right of the main diagonal. Just as diagonal entries are those $A_$ 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:
:$\begin 0 & 2 & 0 \\ 0 & 0 & 3 \\ 0 & 0 & 0 \end$
Likewise, a entry is one that is directly below and to the left of the main diagonal, that is, an entry $A_$ with $j = i - 1$. 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 $A_$ 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.

Antidiagonal

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 $b_$ such that $i + j = N+1$ for all $1 \leq i, j \leq N$. That is, it runs from the top right corner to the bottom left corner.
:$\begin 0 & 0 & \color\\ 0 & \color & 0\\ \color & 0 & 0\end$

(*) ''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.'', $A_$ 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

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Matrices (mathematics)

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