Question

For \( X = (x_1, x_2, x_3)^T \in \mathbb{R}^3 \), consider the quadratic form:

\[ Q(X) = 2x_1^2 + 2x_2^2 + 3x_3^2 + 4x_1x_2 + 2x_1x_3 + 2x_2x_3. \] Let \( M \) be the symmetric matrix associated with the quadratic form \( Q(X) \) with respect to the standard basis of \( \mathbb{R}^3 \). 
Let \( Y = (y_1, y_2, y_3)^T \in \mathbb{R}^3 \) be a non-zero vector, and let

\[ a_n = \frac{Y^T(M + I_3)^{n+1}Y}{Y^T(M + I_3)^n Y}, \quad n = 1, 2, 3, \dots \] Then, the value of \( \lim_{n \to \infty} a_n \) is equal to (in integer).

Hint:

For quadratic forms, find the matrix representation and use the eigenvalues of the matrix to compute limits involving powers of the matrix.

Answer 1

Step 1: Matrix Representation
The quadratic form \( Q(X) \) can be written in matrix form as: 

\[ Q(X) = X^T M X, \] where \( M \) is the symmetric matrix associated with the quadratic form. We need to compute the matrix \( M \) and the limit of the sequence defined by \( a_n \). 

Step 2: Matrix Computation
We compute the matrix \( M \) corresponding to the quadratic form. The matrix \( M \) will be a symmetric matrix with entries corresponding to the coefficients of the quadratic form \( Q(X) \). \[ M = \begin{pmatrix} 2 & 2 & 1 \\ 2 & 2 & 1 \\ 1 & 1 & 3 \end{pmatrix}. \] Step 3: Analyzing the Limit
The expression for \( a_n \) involves powers of the matrix \( M + I_3 \), where \( I_3 \) is the identity matrix. As \( n \to \infty \), the powers of \( M + I_3 \) will be dominated by its largest eigenvalue. Therefore, the ratio \( a_n \) will approach the largest eigenvalue of the matrix \( M + I_3 \). The largest eigenvalue of \( M + I_3 \) is \( 6 \). 
Step 4: Conclusion
Thus, \( \lim_{n \to \infty} a_n = 6 \). 

 \[ \boxed{6} \quad \lim_{n \to \infty} a_n = 6 \]