Word Count : 249

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We are to find the signature of the quadratic form $$ Q(x_1, x_2, x_3) = x_1^2 - 2x_2^2 + 3x_3^2 $$ --- ### Step 1: Write the quadratic form in matrix form A quadratic form in $x_1, x_2, x_3$ can be expressed as: $$ Q(\mathbf{x}) = \mathbf{x}^T A \mathbf{x}, \quad \mathbf{x} = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} $$ Here, the ____ __________ __________ ______ ___ _________ ___ ___ _____ ______.
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