Main diagonal

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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

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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

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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

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Notes

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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)

References

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  • 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.