In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix is the list of entries where . All off-diagonal elements are zero in a diagonal matrix. The following four matrices have their main diagonals indicated by red ones:
Square matrices
[edit]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 with row index specified by and column index specified by , these would be entries with . For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:
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 with , the superdiagonal entries are those with . For example, the non-zero entries of the following matrix all lie in the superdiagonal:
Likewise, a subdiagonal entry is one that is directly below and to the left of the main diagonal, that is, an entry with .[8] General matrix diagonals can be specified by an index measured relative to the main diagonal: the main diagonal has ; the superdiagonal has ; the subdiagonal has ; and in general, the -diagonal consists of the entries with .
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
[edit]The antidiagonal (sometimes counter diagonal, secondary diagonal (*), trailing diagonal, minor diagonal, off diagonal, or bad diagonal) of an order square matrix is the collection of entries such that for all . That is, it runs from the top right corner to the bottom left corner.
(*) 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., 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
[edit]Notes
[edit]- ^ 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)
References
[edit]- 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.