Math Problem Statement
reduce the following quadratic forms into canonical forms by orthogonal transformation3x ^ 2 - 3y ^ 2 - 5z ^ 2 - 2xy - 6yz - 6zx
Solution
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Math Problem Analysis
Mathematical Concepts
Quadratic Forms
Orthogonal Transformation
Matrix Diagonalization
Eigenvalues and Eigenvectors
Formulas
Quadratic form as a matrix: A = [3, -1, -3; -1, -3, -3; -3, -3, -5]
Characteristic equation: det(A - λI) = 0
Eigenvector equation: (A - λI)v = 0
Matrix diagonalization: A = PDP^(-1)
Theorems
Diagonalization Theorem
Spectral Theorem
Suitable Grade Level
College Level (Advanced Linear Algebra)
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