Definition:Pairwise Orthogonal
Jump to navigation
Jump to search
Definition
Let $\sqbrk a_{m n}$ be a matrix of order $m \times n$.
Rows
The rows of $\sqbrk a_{m n}$ are described as pairwise orthogonal if and only if:
- $\forall i, j \in \set {1, 2, \ldots, m}, i \ne j: {r_i}^\intercal \cdot {r_j}^\intercal = 0$
That is, the dot product of each pair of distinct rows of $\sqbrk a_{m n}$, when transposed and considered as vectors, is zero.
Columns
The columns of $\sqbrk a_{m n}$ are described as pairwise orthogonal if and only if:
- $\forall i, j \in \set {1, 2, \ldots, n}, i \ne j: c_i \cdot c_j = 0$
That is, the dot product of each pair of distinct columns of $\sqbrk a_{m n}$, when considered as vectors, is zero.
Also known as
Some sources use the term mutually orthogonal to mean the same thing as pairwise orthogonal.
Also see
- Results about pairwise orthogonality can be found here.