Q: Rank and Full Rank of a Matrix?

Question

I have a couple questions regarding the relationship between the rank of a matrix and the full rank.

Given: Matrix $A$ is rank deficient if it does not have full rank

Definition:

full rank: A matrix $A$ is full rank if it has its maximum possible rank

rank: $\text{Rank}(A)$ is the largest number of columns of $A$ that constitute a linearly independent set

Fact: $\text{Rank}(A) = \text{Rank}(A^T)$

Questions:

A rank deficient matrix $A$ always has a $\text{Rank}(A)$ $\leq$ the full rank correct?

Since column rank = row rank, a non square matrix (2x3, for example) should return a rank $\leq$ 2?

Why is full rank = $\min(m,n)$ where $m$ and $n$ are the number of rows and columns of matrix $A$?

My question here is: why is it that the rank of a matrix can simply be related to the rows/cols instead of analyzing if those rows/cols constitute a linearly independent set?

Thanks!

Answer 1

A rank deficient matrix A always has a Rank(A) ≤ the full rank correct?

Not quite. You need Rank(A)< the full rank. This is just the definition of a rank deficient matrix.

Since column rank = row rank, a non square matrix (2x3, for example) should return a rank ≤ 2?

Its rank will be at most 2. The rank could also be $0$ or $1$. Here are examples:

Rank Zero:

\begin{bmatrix} 0 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}

Rank One: \begin{bmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \end{bmatrix}

Rank Two:

\begin{bmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \end{bmatrix}

The last matrix is an example of a $2 \times 3$ matrix of full rank. The others don't have full rank.

My question here is: why is it that the rank of a matrix can simply be related to the rows/cols instead of analyzing if those rows/cols constitute a linearly independent set?

You do have analyze the linear dependence of the columns or rows. Given a random matrix, let's say $4 \times 7$, it could have rank anywhere between $0$ and $4$, inclusive. If we determine it has rank $4$, we say it has full rank. If we determine it has rank $0, 1, 2,$ or $3$, we say the matrix is rank deficient.

I highly reccomend that you read chapters 1, 2, and 4 in David C. Lay's Linear Algebra and Applications.

Answer 2

$\text{Rank}(A)$ is the largest number of columns (or rows) of $A$ that constitute a linearly independent set.

So if the matrix has $m$ rows, then ofcourse you can't find $m+1$ independent rows for example!