Matrix Rank: Computation, Characterizations, and Applications

All flashcards

What is the rank of a matrix?

The dimension of its column space, row space, or associated linear map's image.

When does a matrix have rank zero?

Exactly when every entry is zero.

What is full rank for an $m\times n$ matrix?

Rank equal to the maximum possible value, $\min(m,n)$.

How is rank deficiency calculated?

Subtract the matrix's rank from $\min(m,n)$.

Why can numerical rank depend on a tolerance?

Floating-point roundoff can make small values appear zero or nonzero.

Quiz questions

1. Which operation is an elementary row operation that preserves matrix rank?Answer: Multiply a row by a nonzero scalar.
2. For a linear map with domain dimension $n$, which equation expresses the rank–nullity theorem?Answer: $\operatorname{rank}(A)+\dim(\ker f)=n$
3. If an $m\times n$ matrix has rank $r$ and is factored as $A=CR$, what are the dimensions of $C$ and $R$?Answer: $C$ is $m\times r$ and $R$ is $r\times n$.
4. How can the rank of a matrix be characterized using its minors?Answer: It is the largest order of a nonzero minor.
5. For compatible matrices $A$ and $B$, which upper bound always holds?Answer: $\operatorname{rank}(AB)\leq\min(\operatorname{rank}(A),\operatorname{rank}(B))$