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. 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.
A entry is one that is directly above and to the right of the main diagonal. 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 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>. 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