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 $a_{i,j}$ 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:

$$\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$$

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 $A_{ij}$ with $i=j$. For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:

$\left(\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right)$

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 superdiagonal entry is one that is directly above and to the right of the main diagonal. 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:

$\left(\begin{array}{ccc}0& 2& 0\\ 0& 0& 3\\ 0& 0& 0\end{array}\right)$

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 $j=i-1$. 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.

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 $b_{i,j}$ such that $i+j=N+1$ for all $1\le i,j\le N$. That is, it runs from the top right corner to the bottom left corner.

$\left[\begin{array}{ccc}0& 0& 1\\ 0& 1& 0\\ 1& 0& 0\end{array}\right]$

(*) 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., $A_{i,i\pm k}$ 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.

See also

• Trace

References

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