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Magazine Problem 1
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. $$ \left[\begin{array}{rr} 3 & -3 \\ -5 & 1 \end{array}\right] $$
Problem 1
Find the upper-triangular coefficient matrix U and the symmetric coefficient matrix \(A\) of the given quadratic form. $$ 3 x^{2}-6 x y+y^{2} $$
Problem 1
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 \(g\). $$ \begin{aligned} &g(x, y)=-7+\left(3 x^{2}-6 x y+4 y^{2}\right)+ \\ &\left(x^{3}-4 y^{3}\right) \end{aligned} $$
Problem 2
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. $$ \left[\begin{array}{ll} 3 & -3 \\ 4 & -5 \end{array}\right] $$
Problem 2
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 \(g\). $$ \begin{aligned} &g(x, y)=8-\left(2 x^{2}-8 x y+3 y^{2}\right)+ \\ &\left(2 x^{2} y-y^{3}\right) \end{aligned} $$
Problem 2
Find the upper-triangular coefficient matrix U and the symmetric coefficient matrix \(A\) of the given quadratic form. $$ 8 x^{2}+9 x y-3 y^{2} $$
Problem 3
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 \(g\). $$ \begin{aligned} &g(x, y)=4-3 x+\left(2 x^{2}-2 x y+y^{2}\right)+ \\ &\left(2 x^{2} y+y^{3}\right)+\cdots \end{aligned} $$
Problem 3
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. $$ \left[\begin{array}{rr} -3 & 10 \\ -3 & 8 \end{array}\right] $$
Problem 3
Find the upper-triangular coefficient matrix U and the symmetric coefficient matrix \(A\) of the given quadratic form. $$ x^{2}-y^{2}-4 x y+3 x z-8 y z $$
Problem 4
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. $$ \left[\begin{array}{ll} -4 & 9 \\ -2 & 5 \end{array}\right] $$
