Main diagonal

 ( Matrices ) 

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:

\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}

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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:

$\backslash begin\{pmatrix\}$

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.

A 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:

$\backslash begin\{pmatrix\}$

0 & 2 & 0 \\
0 & 0 & 3 \\
0 & 0 & 0
     

\end{pmatrix} Likewise, a 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 Band 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.

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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 $1\; \backslash leq\; i,\; j\; \backslash leq\; N$. 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.

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See also

• Trace

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Notes

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