2 Matrix Algebra 2.6 Inverse of a Matrix 2.8 Similarity Transformations

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 $\alpha_{i}$ such that

{\alpha}_{1}\mathbf{a^{1}}+{\alpha}_{2}\mathbf{a^{2}}+...+{\alpha}_{n}\mathbf{% a^{n}}=\mathbf{0}

(23)

is the set

{\alpha}_{1}={\alpha}_{2}=...={\alpha}_{n}=0

For a more concrete example, consider the following matrix.

\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.

	$\displaystyle\mathbf{{a}^{3}}$	$\displaystyle=\mathbf{{a}^{1}+{a}^{2}}$
	$\displaystyle\mathbf{{a}^{4}}$	$\displaystyle=2\mathbf{{a}^{2}}$

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

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

(24)

Furthermore,

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

(25)

and

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

(26)