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Chapter 3: 2P (page 88)
As in Problem 1, write out in detail in terms of equations like (2.6) for two equations in four unknowns; for four equations in two unknowns.
The form iswhich can be written as:
The form is
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Find the equation of the plane through and perpendicular to both planes in Problem 22.
Find the Eigen values and Eigen vectors of the following matrices. Do some problems by hand to be sure you understand what the process means. Then check your results by computer.
Question: Give numerical examples of: a symmetric matrix; a skew-symmetric matrix; a real matrix; a pure imaginary matrix.
Show that an orthogonal matrix M with all real eigenvalues is symmetric. Hints: Method 1. When the eigenvalues are real, so are the eigenvectors, and the unitary matrix which diagonalizes M is orthogonal. Use (11.27). Method 2. From Problem 46, note that the only real eigenvalues of an orthogonal M are ±1. Thus show that . Remember that M is orthogonal to show that .
Show that the product is a symmetric matrix.
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