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To find the orthogonal canonical reduction of the quadratic form \( Q(x, y, z) = x^2 - 2y^2 + z^2 + 2xy + 6yz \), its principal axes, rank, and signature, we proceed step-by-step. --- ### Step 1: Write the quadratic form in matrix form The quadratic form \( Q(x, y, z) \) can be written as: \[ Q(x, y, z) = \mathbf{x}^T A \mathbf{x}, \] where \( \mathbf{x} = \begin{bmatrix} x \\ y \\ z \end{bmatrix} \) and \( A \) is the symmetric matrix associated with the quadratic form. From the given quadratic form: \[ A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & -2 & 3 \\ 0 & 3 & 1 \end{bmatrix}. \] --- ### Step 2: Find the eigenvalues of \( A \) The eigenvalues of \( A \) are found by solving the characteristic equation: \[ \det(A - \lambda I) = 0, \] where \( I \) is the identity matrix. Substituting \( A \): \[ \det \begin{bmatrix} 1 - \lambda & 1 & 0 \\ 1 & -2 - \lambda & 3 \\ 0 & 3 & 1 - \lambda \end{bmatrix} = 0. \] Expanding the determinant: \[ (1 - \lambda) \det __________ _______ __________ ________ _______ ______ _______ ________ _______.
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