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Created by M. Oki Orlandofrom the Noun Project

Rank

Definition (Rank)

Let T∈L(V,W)T\in\mathscr{L}(V,W) be a Linear Map and let AA be the matrix of TT. We say the rank of AA is the dimension of the image of TT. i.e. rank(A)=dim⁡(Image(T))rank(A)=\dim(\text{Image}(T))If rank(A)=dim⁡(V)∧dim⁡(W)rank(A)=\dim(V)\wedge\dim(W) then AA is said to be full rank.

Remark

Full rank implies that ker⁡(T)={0}\ker(T)=\{ 0 \}hence we have that the following also hold:

  1. TT is Injective by Linear Map is Injective iff Kernel is 0
  2. ∀x∈V\forall x\in VxTAx≠0x^{T}Ax\not=0this result is what helps prove that the controllability gramian being full rank is equivalent to it being Positive Definite.

Linked from

Fixed endpoint problem

Linear Time Invariant Control System

Controllability Normal Form