Hi,

need help on these two Linear Algebra question.

Thank you very much.

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4. Let Q be the quadratic form

Q(x) = 5x^{2} - y^{2} + 2z^{2} + 2xy + 10xz - 4yz

on ℝ^{3}. Find a symmetric matrix A such that Q(x) = x^{T}Ax.

This quadratic form is indenite. Demonstrate this using the values Q(x_{1}) and Q(x_{2}),

for suitably chosen vectors x_{1} and x_{2}.

5. For the below quadratic form:

Q(x_{1}; x_{2}; x_{3}) = 2(x_{1}^{2}+ x_{2}^{2}+ x_{3}^{2} - x_{1}x_{2} + x_{1}x_{3} - x_{2}x_{3})

a) Find the symmetric matrix A such that Q(X) = X^{T}AX.

b) Find an orthogonal matrix P such that the change of variables y = P^{T}x transforms

Q into a quadratic form Q′(* y*) with no cross product term. Write down the new

quadratic form Q′(*y*).

c) Classify Q as positive/negative denite or indenite. Justify the answer.

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