Main diagonal

Last updated July 28, 2024

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:

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

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 $A$ 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:

{\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 $A_{ij}$ 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:

{\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 $A_{ij}$ with $j=i-1$ .^[8] 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 $1\leq i,j\leq N$ . That is, it runs from the top right corner to the bottom left corner.

{\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., $A_{i,\,i\pm k}$ 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.

Notes

↑ Bronson (1970 , p. 2)
↑ Herstein (1964 , p. 239)
↑ Nering (1970 , p. 38)
↑ Herstein (1964 , p. 239)
↑ Nering (1970 , p. 38)
↑ Bronson (1970 , pp. 203, 205)
↑ Herstein (1964 , p. 239)
↑ Cullen (1966 , p. 114)

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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 .

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[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)

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