Question
Find the orthogonal canonical reduction of the quadratic form
x2+y2 + z2 - 2xy - 2xz - 2yz. Also, find its principal axes.
Answer :
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The given quadratic form is: \[ Q(x, y, z) = x^2 + y^2 + z^2 - 2xy - 2xz - 2yz \] ### Step 1: Represent the quadratic form in matrix notation A quadratic form can be represented as: \[ Q(x, y, z) = \mathbf{x}^T A \mathbf{x} \] where \(\mathbf{x} = \begin{pmatrix} x \\ y \\ z \end{pmatrix}\) and \(A\) is a symmetric matrix. We can express the given quadratic form in the matrix form by looking at the coefficients of the terms: \[ Q(x, y, z) = x^2 + y^2 + z^2 - 2xy - 2xz - 2yz \] Thus, the matrix \(A\) is: \[ A = \begin{pmatrix} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{pmatrix} \] ### Step 2: Find the eigenvalues and eigenvectors of \(A\) To diagonalize the matrix \(A\), we need to find its eigenvalues and eigenvectors. The eigenvalues \(\lambda\) are found by solving the characteristic equation: \[ \text{det}(A - \lambda I) = 0 \] where \(I\) is the identity matrix. The determinant of
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