# Rank of a Matrix

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:

V=\begin{bmatrix}v_1\begin{bmatrix}v_1\\v_2\\ \vdots\\v_r \end{bmatrix}&v_2\begin{bmatrix}v_1\\v_2\\ \vdots\\v_r \end{bmatrix}&\dots &v_r\begin{bmatrix}v_1\\v_2\\ \vdots\\v_r \end{bmatrix}\end{bmatrix}

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.