Question

Question: If A is \(m \times n\), then the matrix \(G = {A^T}A\) is called the Gram matrix of A. In this case, the entries of G are the inner products of the columns of A. (See Exercises 9 and 10).

9. Show that the Gram matrix of any matrix A is positive semidefinite, with the same rank as A. (See the Exercises in Section 6.5.)

Short answer

It is proved that the Gram matrix of any matrix \(A\) is positive semidefinite with the same rank as A.

Step-by-step solution

Gram matrix

When \(A\) is a \(m \times n\) matrix then the matrix \(G = {A^T}A\) is known as theGram matrix of A.

Show that the Gram matrix of any matrix A is positive semidefinite, with the same rank as A

Exercise 22 in section 6.5states that \({\mathop{\rm rank}\nolimits} {A^T}A = {\mathop{\rm rank}\nolimits} A\).

When \(A\) is a \(m \times n\) matrix and \({\bf{x}}\) in \({\mathbb{R}^n}\) then;

\(\begin{array}{c}{{\bf{x}}^T}{A^T}A{\bf{x}} = {\left( {A{\bf{x}}} \right)^T}\left( {A{\bf{x}}} \right)\\ = {\left\| {A{\bf{x}}} \right\|^2} \ge 0\end{array}\)

Therefore, \(G = {A^T}A\) is a positive semidefinite. According to Exercise 22 in Section 6.5, \({\mathop{\rm rank}\nolimits} {A^T}A = {\mathop{\rm rank}\nolimits} A\).

Thus, it is proved that the Gram matrix of any matrix \(A\) is positive semidefinite with the same rank as A.