91Ó°ÊÓ

Use the power method to estimate the eigenvalue of maximum magnitude and a corresponding eigenvector for the given matrix. Start with first estimate $$ w_{1}=\left[\begin{array}{l} 1 \\ 1 \end{array}\right] \text {, } $$ and compute \(\mathrm{w}_{2}, \mathrm{w}_{3}\), and \(\mathrm{w}_{4}\). Also find the three Rayleigh quotients. Then find the exact eigenvalues, for comparison, using the characteristic equation. $$ \left[\begin{array}{ll} -4 & 9 \\ -2 & 5 \end{array}\right] $$

Find the upper-triangular coefficient matrix U and the symmetric coefficient matrix \(A\) of the given quadratic form. $$ \begin{aligned} &x_{1}{ }^{2}-2 x_{2}{ }^{2}+x_{3}{ }^{2}+6 x_{4}{ }^{2}-2 x_{1} x_{4}+ \\ &6 x_{2} x_{4}-8 x_{1} x_{3} \end{aligned} $$

Find the spectral decomposition (8) of the given symmetric matrix. $$ \left[\begin{array}{ll} 2 & 3 \\ 3 & 2 \end{array}\right] $$

Find the upper-triangular coefficient matrix U and the symmetric coefficient matrix \(A\) of the given quadratic form. $$ [x, y]\left[\begin{array}{rr} 7 & -10 \\ 15 & 2 \end{array}\right]\left[\begin{array}{l} x \\ y \end{array}\right] $$

Assume that \(g(\mathbf{x})\), or \(g(\mathbf{x})\), is described by the given formula for values of \(\mathbf{x}\), or \(\overline{\mathbf{x}}\), near zero. Draw whatever conclusions are possible concerning local extrema of the function \(g\). $$ \begin{aligned} &g(x, y)=-2+\left(8 x^{2}+4 x y+y^{2}\right)+ \\ &\left(x^{3}+5 x^{2} y\right) \end{aligned} $$

Find the spectral decomposition (8) of the given symmetric matrix. $$ \left[\begin{array}{ll} 3 & 5 \\ 5 & 3 \end{array}\right] $$

Find the spectral decomposition (8) of the given symmetric matrix. $$ \left[\begin{array}{lll} 1 & 1 & 1 \\ 1 & 0 & 0 \\ 1 & 0 & 0 \end{array}\right] $$

Find the upper-triangular coefficient matrix U and the symmetric coefficient matrix \(A\) of the given quadratic form. $$ [x, y, z]\left[\begin{array}{rrr} 8 & 3 & 1 \\ 2 & 1 & -4 \\ -5 & 2 & 10 \end{array}\right]\left[\begin{array}{l} x \\ y \\ z \end{array}\right] $$

Assume that \(g(\mathbf{x})\), or \(g(\mathbf{x})\), is described by the given formula for values of \(\mathbf{x}\), or \(\overline{\mathbf{x}}\), near zero. Draw whatever conclusions are possible concerning local extrema of the function \(g\). $$ \begin{aligned} &g(x, y)=5+\left(3 x^{2}+10 x y+7 y^{2}\right)+ \\ &\left(7 x y^{2}-y^{3}\right) \end{aligned} $$

Find the spectral decomposition (8) of the given symmetric matrix. $$ \left[\begin{array}{rrr} 1 & -1 & -1 \\ -1 & 1 & -1 \\ -1 & -1 & 1 \end{array}\right] $$