Final answer:
To reduce the given quadratic form to canonical form using orthogonal transformation, an orthogonal matrix that diagonalizes the associated symmetric matrix is required. This involves finding eigenvalues and eigenvectors of the matrix, orthogonalizing and normalizing them to form the matrix, and finally applying the transformation.
Step-by-step explanation:
To reduce the quadratic form 2t_1^2 + 6t_1t_2 + 5t_2^2 - 2t_2t_3 + 2t_3^2 to canonical form using orthogonal transformation, we must find an orthogonal matrix that diagonalizes the associated symmetric matrix of the quadratic form. The process involves using the eigenvalue-eigenvector method to determine the orthogonal matrix.
First, we write the quadratic form as a matrix:
[2 3 0]
[3 5 -1]
[0 -1 2]
Next, we find the eigenvalues by solving the characteristic equation of the matrix. Once the eigenvalues are found, we calculate the corresponding eigenvectors. These eigenvectors are then orthogonalized and normalized to form the columns of the orthogonal matrix.
Then, by performing the transformation t' = P^{-1}t, where P is the orthogonal matrix and t is the original variable vector, we obtain the new variable vector t'. The transformed quadratic form in terms of t' will have the coefficients as the eigenvalues on the diagonal and zeros elsewhere, which is the canonical form.
The process of reducing a quadratic form to canonical form is closely related to the principal axis theorem and often involves complex algebra and matrix computations.