Science:Math Exam Resources/Courses/MATH307/April 2005/Question 03 (c)

MATH307 April 2005

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Question 03 (c)
Let P be a real symmetric n x n matrix such that P2 = P. Also let R = In - 2P be the matrix that reflects a vector across a plane. (c) Find the eigenvalues of R and give a description of the corresponding eigenspaces.

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Hint

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Geometrically we can see that the eigenvalues of a reflection matrix are$-1$ (in the case that they reflect over something other than itself) or $1$ if it reflects over itself. The corresponding eigenspace associating to $\lambda _{1}=-1$ is the line normal, and the eigenspace associating to $\lambda _{2}=1$ is the line of reflection.