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    Gram matrix of the Example 2 inner product

    E-HK-81-10

    Area
    Algebra
    Topics
    Inner Product Spaces, Linear Algebra
    Status
    draft

    In Example 2 the inner product on \(\mathbb R^2\) is \[ ((x_1,x_2)\mid(y_1,y_2)) =x_1y_1-x_2y_1-x_1y_2+4x_2y_2. \] Let \(B\) be the standard ordered basis for \(\RR^2\). Find the matrix of this inner product relative to \(B\).

    Solution

    Let \[ e_1=(1,0),\qquad e_2=(0,1). \] The Gram matrix relative to \(B=(e_1,e_2)\) is \[ G=\bigl((e_i\mid e_j)\bigr)_{i,j=1}^2. \] Directly from the defining formula, \[ (e_1\mid e_1)=1, \qquad (e_1\mid e_2)=-1, \qquad (e_2\mid e_1)=-1, \qquad (e_2\mid e_2)=4. \] Therefore \[ \boxed{ G=\begin{pmatrix}1&-1\\-1&4\end{pmatrix}.} \]

    Indeed, if \(X=(x_1,x_2)^t\) and \(Y=(y_1,y_2)^t\), then \[ X^tGY =x_1y_1-x_1y_2-x_2y_1+4x_2y_2, \] which recovers the given inner product exactly [@HK71].

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