2.7 Rank of a Matrix

The rank of an n × n matrix A is the maximum number of linearly independent columns in A. Column vectors of of A, denoted ai, are linearly independent if the only set of scalars αi\alpha_{i} such that

α1𝐚𝟏+α2𝐚𝟐++αn𝐚𝐧=𝟎{\alpha}_{1}\mathbf{a^{1}}+{\alpha}_{2}\mathbf{a^{2}}+...+{\alpha}_{n}\mathbf{% a^{n}}=\mathbf{0} (7)

is the set

α1=α2==αn=0{\alpha}_{1}={\alpha}_{2}=...={\alpha}_{n}=0

For a more concrete example, consider the following matrix.

𝐀=(011212342020)\mathbf{A}=\left(\begin{array}[]{cccc}0&1&1&2\\ 1&2&3&4\\ 2&0&2&0\end{array}\right)

The rank of A is two, since its third and fourth columns are linear combinations of its first two columns, i.e.

𝐚𝟑\displaystyle\mathbf{{a}^{3}} =𝐚𝟏+𝐚𝟐\displaystyle=\mathbf{{a}^{1}+{a}^{2}}
𝐚𝟒\displaystyle\mathbf{{a}^{4}} =2𝐚𝟐\displaystyle=2\mathbf{{a}^{2}}

If A is an n × n matrix, it can be shown

rank(𝐀)min(m,n)\text{rank}\left(\mathbf{A}\right)\leq\text{min}(m,n) (8)

Furthermore,

rank(𝐀𝐁) min(rank(𝐀),rank(𝐁))\text{rank}\left(\mathbf{AB}\right)\leq\text{ min}(\text{rank}\left(\mathbf{A}% \right),\text{rank}\left(\mathbf{B}\right)) (9)

and

rank(𝐀𝐀𝐓)=rank(𝐀𝐓𝐀)=rank(𝐀)\text{rank}\left(\mathbf{A{A}^{T}}\right)=\text{rank}\left(\mathbf{{A}^{T}A}% \right)=\text{rank}\left(\mathbf{A}\right) (10)