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Cholesky factorization for $\mathbf{2}\mathbf{×}\mathbf{2}$ matrices. Show that any positive definite $\mathbf{2}\mathbf{×}\mathbf{2}$ matrix A can be written uniquely $\mathit{A}\mathbf{=}\mathit{L}{\mathit{L}}^{\mathbf{T}}$ as where L is a lower triangular$\mathbf{2}\mathbf{×}\mathbf{2}$ matrix with positive entries on the diagonal. Hint: Solve the equation

$\left[\begin{array}{cc}a& b\\ b& c\end{array}\right]\mathbf{=}\left[\begin{array}{cc}x& 0\\ y& z\end{array}\right]\left[\begin{array}{c}\begin{array}{cc}x& y\\ 0& z\end{array}\end{array}\right]$

37. If Ais a positive definite$\mathbf{n}\mathbf{×}\mathbf{n}$matrix and$\stackrel{\mathbf{→}}{\mathbf{x}}$ is a nonzero vector in${\mathbf{R}}^{\mathbf{n}}$, then the angle between$\stackrel{\mathbf{→}}{\mathbf{x}}$and$\mathbf{A}\stackrel{\mathbf{→}}{\mathbf{x}}$must be acute.

Find a symmetric 2x2matrix Bsuch that${\mathit{B}}^{\mathbf{3}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{5}}\left[\begin{array}{cc}12& 14\\ 14& 33\end{array}\right]$

Find the singular values of $\mathit{A}\mathbf{=}\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$. Find a unit vector$\stackrel{\mathbf{→}}{\mathbf{v}}$such that$||A\stackrel{→}{v_1}||\mathbf{=}{\mathbf{σ}}_{\mathbf{1}}$. Sketch the image of the unit circle.

Show that any positive definite n x n matrix A can be written as A=BB^T, where B is a n x n matrix with orthogonal columns. Hint: There exists an orthogonal matrix S such that S^-1AS = S^TAS = D is a diagonal matrix with positive diagonal entries. Then A=SDS^T. Now write D as the square of a diagonal matrix.