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Chapter 3: Q34P (page 160)
By multiplying out where C is the rotation matrix (11.14) and D is the diagonal matrix
Show that if M can be diagonalized by a rotation, then M is symmetric.
can be diagonalized by a rotation and is symmetric.
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Write the matrices which produce a rotation about the axis, or that rotation combined with a reflection through the (y,z) plane.
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.
Are the following linear vector functions? Prove your conclusions using (7.2).
4.,whereAis a given vector.
Let each of the following matricesM describe a deformation of the ( x , y)plane for each given Mfind: the Eigen values and eigenvectors of the transformation, the matrix Cwhich Diagonalizes Mand specifies the rotation to new axesalong the eigenvectors, and the matrix D which gives the deformation relative to the new axes. Describe the deformation relative to the new axes.
Question: For each of the following problems write and row reduce the augmented matrix to find out whether the given set of equations has exactly one solution, no solutions, or an infinite set of solutions. Check your results by computer. Warning hint:Be sure your equations are written in standard form. Comment: Remember that the point of doing these problems is not just to get an answer (which your computer will give you), but to become familiar with the terminology, ideas, and notation we are using
6.
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