18.10 Rank of a Matrix

Types: N/A
Examples: N/A
Constructions: N/A
Generalizations: 7.1 Entry-by-Entry Definition of Matrix

Properties: N/A
Sufficiencies: N/A
Questions: N/A

Rank of a Matrix

Let $A \in R^{m \times n}$ be a "given" matrix. The rank of our matrix $A$ is the number of pivots. To refer to this number, we define the nonnegative integer

r = rank (A)

which counts:

the number of pivots in $A$ .
the number of pivot columns in $A$ .
the number of linearly independent columns in $A$ .
the number of pivot rows in $A$ .
the number of linearly independent rows in $A$ .

The number of pivot columns equals the number of pivot rows.

Remark. The pivot columns of $A$ are identical to the pivot columns of $U$ , but the pivot rows of $A$ are NOT identical to the pivot rows of $U$ because when we matrix multiply on the left, we manipulate the row space of matrix $A$ . However, the count is the same.
$⧫$

Bounds for the Rank of a Matrix

Let $A \in R^{m \times n}$ be a "given matrix."

There can be no more than one pivot per row.
There can be no more than one pivot per column.

Thus, the rank of the $m \times n$ matrix $A$ is bounded above:

0 \leq r = rank (A) \leq min (m, n)

In other words, if we have a short wide matrix, the rank is bounded by the row dimension. If we have a tall thin matrix, the rank is bounded by the column dimension.

Remark. The only matrix with a zero rank is the zero matrix.
$⧫$