Definition:Matrix/Diagonal
Definition
Let $\mathbf A$ be a matrix.
A diagonal of $\mathbf A$ is a diagonal line of elements of $\mathbf A$.
Main Diagonal
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.
The elements $a_{j j}: j \in \closedint 1 {\min \set {m, n} }$ constitute the main diagonal of $\mathbf A$.
That is, the main diagonal of $\mathbf A$ is the diagonal of $\mathbf A$ from the top left corner, that is, the element $a_{1 1}$, running towards the lower right corner.
Main Antidiagonal
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.
The main antidiagonal of $\mathbf A$ is the antidiagonal of $\mathbf A$ from the top right corner, that is, the element $a_{1 n}$, running towards the lower left corner.
Superdiagonal
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.
The superdiagonals of $A$ are the diagonals of $\mathbf A$ lying parallel to and above the main diagonal of $\mathbf A$.
That is, the elements $\map a {r + k, s + k}$ where $s > r$.
Subdiagonal
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.
The subdiagonals of $A$ are the diagonals of $\mathbf A$ lying parallel to and below the main diagonal of $\mathbf A$.
That is, the elements $\map a {r + k, s + k}$ where $s < r$.
Antidiagonal
Let $\mathbf A = \sqbrk a_{m n}$ be a matrix.
An antidiagonal of $A$ is a diagonal of $\mathbf A$ lying perpendicular to the main diagonal of $\mathbf A$.
That is, a set of elements $\map a {r + k, s - k}$.
Also defined as
Some sources define a diagonal of a matrix for a square matrix only.
Also see
- Results about matrix diagonals can be found here.
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): matrix (plural matrices)
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): matrix (plural matrices)