Main diagonal

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.<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref> 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.<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref>

A superdiagonal entry is one that is directly above and to the right of the main diagonal.<ref>Template:Harvtxt</ref><ref>Template:Harvtxt</ref> 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 subdiagonal 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>.<ref>Template:Harvtxt</ref> 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

References

• Template:Citation
• Template:Citation
• Template:Citation
• Template:Citation
• {{#invoke:Template wrapper|{{#if:|list|wrap}}|_template=cite web

|_exclude=urlname, _debug, id |url = https://mathworld.wolfram.com/{{#if:Diagonal%7CDiagonal.html}} |title = Main diagonal |author = Weisstein, Eric W. |website = MathWorld |access-date = |ref = Template:SfnRef }}

Template:Matrix-stub