Main diagonal

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

$${\displaystyle {\begin{bmatrix}\color {red}{1}&0&0\\0&\color {red}{1}&0\\0&0&\color {red}{1}\end{bmatrix}}\qquad {\begin{bmatrix}\color {red}{1}&0&0&0\\0&\color {red}{1}&0&0\\0&0&\color {red}{1}&0\end{bmatrix}}\qquad {\begin{bmatrix}\color {red}{1}&0&0\\0&\color {red}{1}&0\\0&0&\color {red}{1}\\0&0&0\end{bmatrix}}\qquad {\begin{bmatrix}\color {red}{1}&0&0&0\\0&\color {red}{1}&0&0\\0&0&\color {red}{1}&0\\0&0&0&\color {red}{1}\end{bmatrix}}}$$

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. ^{[1] [2] [3]} For a matrix ${\displaystyle A}$ with row index specified by ${\displaystyle i}$ and column index specified by ${\displaystyle j}$, these would be entries ${\displaystyle A_{ij}}$ with ${\displaystyle i=j}$. For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:

${\displaystyle {\begin{pmatrix}1&0&0\\0&1&0\\0&0&1\end{pmatrix}}}$

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. ^{[4] [5]}

A superdiagonal entry is one that is directly above and to the right of the main diagonal. ^{[6] [7]} Just as diagonal entries are those ${\displaystyle A_{ij}}$ with ${\displaystyle j=i}$, the superdiagonal entries are those with ${\displaystyle j=i+1}$. For example, the non-zero entries of the following matrix all lie in the superdiagonal:

${\displaystyle {\begin{pmatrix}0&2&0\\0&0&3\\0&0&0\end{pmatrix}}}$

Likewise, a subdiagonal entry is one that is directly below and to the left of the main diagonal, that is, an entry ${\displaystyle A_{ij}}$ with ${\displaystyle j=i-1}$. ^[8] General matrix diagonals can be specified by an index ${\displaystyle k}$ measured relative to the main diagonal: the main diagonal has ${\displaystyle k=0}$; the superdiagonal has ${\displaystyle k=1}$; the subdiagonal has ${\displaystyle k=-1}$; and in general, the ${\displaystyle k}$-diagonal consists of the entries ${\displaystyle A_{ij}}$ with ${\displaystyle j=i+k}$.

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

See also: Anti-diagonal matrix

The antidiagonal (sometimes counter diagonal, secondary diagonal (*), trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order ${\displaystyle N}$ square matrix ${\displaystyle B}$ is the collection of entries ${\displaystyle b_{i,j}}$ such that ${\displaystyle i+j=N+1}$ for all ${\displaystyle 1\leq i,j\leq N}$. That is, it runs from the top right corner to the bottom left corner.

${\displaystyle {\begin{bmatrix}0&0&\color {red}{1}\\0&\color {red}{1}&0\\\color {red}{1}&0&0\end{bmatrix}}}$

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

Notes

1. Bronson (1970 , p. 2)
2. Herstein (1964 , p. 239)
3. Nering (1970 , p. 38)
4. Herstein (1964 , p. 239)
5. Nering (1970 , p. 38)
6. Bronson (1970 , pp. 203, 205)
7. Herstein (1964 , p. 239)
8. Cullen (1966 , p. 114)

References

• Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN   70097490
• Cullen, Charles G. (1966), Matrices and Linear Transformations, Reading: Addison-Wesley, LCCN   66021267
• Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN   978-1114541016
• Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN   76091646
• Weisstein, Eric W. "Main diagonal". MathWorld .