(See Steps) In this problem we have the symmetric matrix
Question: In this problem we have the symmetric matrix
\[A=\left(\begin{array}{lll} 3 & 0 & 4 \\ 0 & 2 & 0 \\ 4 & 0 & 3 \end{array}\right)\]- Calculate the characteristic polynomial \(\operatorname{det}\left(A-\lambda I_{3}\right)\).
- Find the eigenvalues of \(A\).
- For each eigenvalue of \(A\), find an orthonormal basis for the corresponding eigenspace.
- Find an orthogonal matrix \(P\) and a diagonal matrix \(D\) so that \(A=P D P^{t}\).
- Show that \(A^{n}=P D^{n} P^{t}\) for each integer \(n\). In particular, calculate \(A^{3}\).
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Show that \(A-\alpha I_{3}=P\left(D-\alpha I_{3}\right) P^{t}\) for each \(\alpha \in \mathbb{R}\). Then calculate \(\left(A-3 I_{3}\right)^{-1}\).
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(See Solution) Use the Lagrange multiplier method to find the extrema of
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(See Steps) Let f(x, y)=x^2+y^2+x y on the unit disk D=(x, y)
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(Solution Library) Find the shortest distance from the origin
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[Solution] Consider the quadratic form Q(x, y, z)=3 x^2+8 x z+2
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(Step-by-Step) Suppose that
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![[Solution] Consider the quadratic form Q(x, y,](/images/solutions/MC-solution-library-72163.jpg "[Solution] Consider the quadratic form Q(x, y, z)=3 x^2+8 x z+2")
