91Ó°ÊÓ

Determine the definiteness of the quadratic forms. $$q\left(x_{1}, x_{2}\right)=x_{1}^{2}+4 x_{1} x_{2}+x_{2}^{2}$$

Find the singular values of \(A=\left[\begin{array}{rr}p & -q \\ q & p\end{array}\right] .\) Explain your answer geometrically.

For each of the matrices in Exercises 1 through \(6,\) find an orthonormal eigenbasis. Do not use technology. $$\left[\begin{array}{rrr}0 & 2 & 2 \\\2 & 1 & 0 \\\2 & 0 & -1\end{array}\right]$$

Determine the definiteness of the quadratic forms. $$q\left(x_{1}, x_{2}\right)=2 x_{1}^{2}+6 x_{1} x_{2}+4 x_{2}^{2}$$

Find the singular values of \(A=\left[\begin{array}{ll}1 & 2 \\ 2 & 4\end{array}\right] .\) Find a unit vector \(\vec{v}_{1}\) such that \(\left\|A \vec{v}_{1}\right\|=\sigma_{1} .\) Sketch the image of the unit circle.

For each of the matrices \(A\) in Exercises 7 through \(11,\) find an orthogonal matrix S and a diagonal matrix \(D\) such that \(S^{-1} A S=D .\) Do not use technology. $$A=\left[\begin{array}{ll}3 & 2 \\\2 & 3\end{array}\right]$$

For each of the matrices \(A\) in Exercises 7 through \(11,\) find an orthogonal matrix S and a diagonal matrix \(D\) such that \(S^{-1} A S=D .\) Do not use technology. $$A=\left[\begin{array}{rr}3 & 3 \\\3 & -5\end{array}\right]$$

Recall that a real square matrix \(A\) is called skew symmetric if \(A^{T}=-A\) a. If \(A\) is skew symmetric, is \(A^{2}\) skew symmetric as well? Or is \(A^{2}\) symmetric? b. If \(A\) is skew symmetric, what can you say about the definiteness of \(A^{2} ?\) What about the eigenvalues of \(A^{2} ?\) c. What can you say about the complex eigenvalues of a skew-symmetric matrix? Which skew-symmetric matrices are diagonalizable over \(\mathbb{R} ?\)

For each of the matrices \(A\) in Exercises 7 through \(11,\) find an orthogonal matrix S and a diagonal matrix \(D\) such that \(S^{-1} A S=D .\) Do not use technology. $$A=\left[\begin{array}{lll}0 & 0 & 3 \\\0 & 2 & 0 \\\3 & 0 & 0\end{array}\right]$$

For each of the matrices \(A\) in Exercises 7 through \(11,\) find an orthogonal matrix S and a diagonal matrix \(D\) such that \(S^{-1} A S=D .\) Do not use technology. $$A=\left[\begin{array}{rrr}1 & -2 & 2 \\\\-2 & 4 & -4 \\\2 & -4 & 4\end{array}\right]$$