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) = 5x² - y² + 2z² + 2xy + 10xz - 4yz

on ³. Find a symmetric matrix A such that Q(x) = x^TAx.

This quadratic form is indenite. Demonstrate this using the values Q(x₁) and Q(x₂),

for suitably chosen vectors x₁ and x₂.

5. For the below quadratic form:

Q(x₁; x₂; x₃) = 2(x₁²+ x₂²+ x₃² - x₁x₂ + x₁x₃ - x₂x₃)

a) Find the symmetric matrix A such that Q(X) = X^TAX.

b) Find an orthogonal matrix P such that the change of variables y = P^Tx 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.