Dyad

Description:

• $A\in {\mathbb{R}}^{n\times m}$ such that in can be written as $A=p{q}^T$ for some vectors $p\in {\mathbb{R}}^n,q\in {\mathbb{R}}^m$
  • outer product of 2 vector
  • The columns of $A$ are scaled copies the same column $p$, with scaling factors given in vector $q$
  • The rows of $A$ are scaled copies the same row $q^{⊺}$, with scaling factors given in vector $p$
• $A(i,j)=u(i).v(j),1\le i\le n,1\le j\le m$
  • ex: differentiation
  • 
• $Ax=(u{v}^T)x=(v^{T}x)u$
  • In terms of the associated linear map, for a dyad, the output always points in the same direction u in output space $({\mathbb{R}}^m)$, no matter what the input x is.
  • The output is thus always a simple scaled version of u.
  • The amount of scaling depends on the vector v, via the linear function $x\to v^{T}x$
• Having $k$ independent columns and $k$ small compared to $m,n$

Dyads in low rank matrix:

• For $k<<m,n$ a matrix $m\times n$ with rank-k is in the form $A=U{V}^{⊺}$, for some matrices $U,V$ with $k$ independent column each. That is $U\in {\mathbb{R}}^{n\times k}$ and $V\in {\mathbb{R}}^{m\times k}$
• Then $A$ has the form of sum of $k$ dyads: $A=(u_1...u_k)\left(\begin{array}{c}{v}_1^{⊺}\\ .\\ .\\ v_k^{⊺}\end{array}\right)=\sum _{i=1}^k{u}_i{v}_i^{⊺}$
• Its elements are given by: $A(i,j)={\sum }_{l=1}^k{u}_l(i)v_l(j)$