Definition:Pairwise Orthogonal

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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.