In linear algebra, the main diagonal (sometimes principal diagonal, primary diagonal, leading diagonal, major diagonal, or good diagonal) of a matrix A {\displaystyle A} is the list of entries a i , j {\displaystyle a_{i,j}} where i = j {\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:
[ 1 0 0 0 1 0 0 0 1 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 ] [ 1 0 0 0 1 0 0 0 1 0 0 0 ] [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] {\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.123 For a matrix A {\displaystyle A} with row index specified by i {\displaystyle i} and column index specified by j {\displaystyle j} , these would be entries A i j {\displaystyle A_{ij}} with i = j {\displaystyle i=j} . For example, the identity matrix can be defined as having entries of 1 on the main diagonal and zeroes elsewhere:
( 1 0 0 0 1 0 0 0 1 ) {\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.45
A superdiagonal entry is one that is directly above and to the right of the main diagonal.67 Just as diagonal entries are those A i j {\displaystyle A_{ij}} with j = i {\displaystyle j=i} , the superdiagonal entries are those with j = i + 1 {\displaystyle j=i+1} . For example, the non-zero entries of the following matrix all lie in the superdiagonal:
( 0 2 0 0 0 3 0 0 0 ) {\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 A i j {\displaystyle A_{ij}} with j = i − 1 {\displaystyle j=i-1} .8 General matrix diagonals can be specified by an index k {\displaystyle k} measured relative to the main diagonal: the main diagonal has k = 0 {\displaystyle k=0} ; the superdiagonal has k = 1 {\displaystyle k=1} ; the subdiagonal has k = − 1 {\displaystyle k=-1} ; and in general, the k {\displaystyle k} -diagonal consists of the entries A i j {\displaystyle A_{ij}} with j = i + k {\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 N {\displaystyle N} square matrix B {\displaystyle B} is the collection of entries b i , j {\displaystyle b_{i,j}} such that i + j = N + 1 {\displaystyle i+j=N+1} for all 1 ≤ i , j ≤ N {\displaystyle 1\leq i,j\leq N} . That is, it runs from the top right corner to the bottom left corner.
[ 0 0 1 0 1 0 1 0 0 ] {\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., A i , i ± k {\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
Notes
- 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 {{citation}}: ISBN / Date incompatibility (help)
- Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646
- Weisstein, Eric W. "Main diagonal". MathWorld.
References
Bronson (1970, p. 2) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 https://lccn.loc.gov/70097490 ↩
Herstein (1964, p. 239) - Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 ↩
Nering (1970, p. 38) - Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 https://lccn.loc.gov/76091646 ↩
Herstein (1964, p. 239) - Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 ↩
Nering (1970, p. 38) - Nering, Evar D. (1970), Linear Algebra and Matrix Theory (2nd ed.), New York: Wiley, LCCN 76091646 https://lccn.loc.gov/76091646 ↩
Bronson (1970, pp. 203, 205) - Bronson, Richard (1970), Matrix Methods: An Introduction, New York: Academic Press, LCCN 70097490 https://lccn.loc.gov/70097490 ↩
Herstein (1964, p. 239) - Herstein, I. N. (1964), Topics In Algebra, Waltham: Blaisdell Publishing Company, ISBN 978-1114541016 ↩
Cullen (1966, p. 114) - Cullen, Charles G. (1966), Matrices and Linear Transformations, Reading: Addison-Wesley, LCCN 66021267 https://lccn.loc.gov/66021267 ↩