Rank of (A B) & (A + B)

Rank inequalities and equalities for matrix products and sums, visual and geometric interpretations of rank loss, multiplication by invertible matrices, rank(AᵀA) = rank(A), and comparison between AB and A + B.

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Topic	Pages
rank(AB) ≤ rank(B) & rank(AB) = rank(B) − dim( col(B) ∩ null(A) )	1–3
Visual illustration of rank(AB) = rank(B) − dim( col(B) ∩ null(A) )	3–4
Extension of rank(AB) = rank(B) − dim( col(B) ∩ null(A) )	5–6
rank(AB) ≤ rank(A) & rank(AB) = rank(A) − dim( row(A) ∩ ℓ-null(B) )	6–8
Multiplication by invertible matrix	8
Geometric view of rank(AB)	9
Geometric view of rank(AB) < min (rank(A), rank(B))	10
rank(AᵀA) = rank(A)	10–11
rank(A + B) ≤ rank(A) + rank(B) − dim( col(A) ∩ col(B) )	12–13
Comparison between AB & A + B	13–14

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