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Rank of a Product

How does multiplication affect rank? Two fundamental inequalities: $\text{Rank}(AB) \leq \text{Rank}(A)$ and $\text{Rank}(AB) \leq \text{Rank}(B)$ .

Geometrically: $\text{Col}(AB) \subseteq \text{Col}(A)$ — the outputs of $AB$ are also outputs of $A$ , so $AB$ can't exceed $A$ 's rank. Similarly, $\text{Null}(AB) \supseteq \text{Null}(B)$ — anything that $B$ kills, $AB$ also kills.

If either $A$ or $B$ has full rank (is injective or surjective respectively), then multiplying by it doesn't reduce rank. In particular, multiplying by an invertible matrix preserves rank.

Formal View

Theorem 4.7 (Rank Inequalities)

For

A \in \mathbb{R}^{m \times n}

and

B \in \mathbb{R}^{n \times p}

: \begin{enumerate} \item

\text{Col}(AB) \subseteq \text{Col}(A)

, so

\text{Rank}(AB) \leq \text{Rank}(A)

\item

\text{Null}(B) \subseteq \text{Null}(AB)

, so

\text{Nullity}(AB) \geq \text{Nullity}(B)

\item

\text{Rank}(AB) \leq \min(\text{Rank}(A), \text{Rank}(B))

\end{enumerate}

Why This Matters

Rank inequalities for products appear throughout matrix analysis and algorithm design.

Low-rank matrix approximations: if $A = BC$ with $B, C$ low-rank, then $A$ is low-rank
In neural networks, weight matrices at each layer bound the effective rank of learned representations
PCA: the product of a data matrix with its transpose has rank = number of principal components

Learning Resources

Rank of matrix products

MIT OpenCourseWare

Rank inequalities and the column space inclusion.

49 min

Quiz

Question 1

If Rank( $A$ ) = 2 and Rank( $B$ ) = 5, what can we say about Rank( $AB$ )?

Common Mistakes

Thinking rank is multiplicative — Rank( $AB$ ) $\leq$ min(Rank $A$ , Rank $B$ ), not equal.