Let (the results also holds for ). Then, the **column rank**/**row rank** of A is defined as the dimension of the column/row space of A, i.e. the dimension of the vector space spanned by the columns/rows of A; this is then equivalent to the number of linearly independent columns/rows (column/rows vector) of A.

**Theorem:** column rank of A = row rank of A.

**Definition:** the rank of a matrix, **rank(****A****)**, is the dimension of either the column or row space of A; simply the number of linearly independent columns or rows of A.

**Definition:** for a linear map , the rank of the linear map is defined as the dimension of the image of . This definition is equivalent to the definition of the matrix rank as every linear map has a matrix by which it can be written as .

**Proposition**: . This leads to these definitions: A matrix is said to be **full rank** iff , i.e. the largest possible rank, and it is said to be **rank deficient** iff , i.e. not having full rank.

## Properties of rank

For :

**1-** only a zero matrix has rank zero.

**2-** If , then

**3-**

**4-**

**5-**

**6-** If , then for , . In addition, for , , i.e. has at most rank *n*.

**7-** A square matrix can be decomposed as where is a diagonal matrix containing the eigenvalues of . Then, , i.e. the number of non-zero eigenvalues of .

**8-** For a square matrix , then equivalently is full rank, is invertible, has non-zero determinant, and has *n* non-zero eigenvalues.

## Proofs

**P6:**

where are coordinates of the vector . This indicates that each column of is a scalar multiple of any other columns of ; therefore, the column space is one dimensional. Hence, .

For the second part, property 3 proves the statement.