Example 1. Let π be a plane through the origin and let S be the transformation which reflects each vector through π. If Y is a vector on π, then S(Y) = Y, and if U is a vector perpendicular to π, then S(U) = −U. Thus for t = 1 and t = −1, there exist nonzero vectors X satisfying S(X) = tX. If X is any vector which is neither on π nor perpendicular to π, then S(X) is not a multiple of X.
KeywordsCharacteristic Equation Linear Transformation Linear Algebra Real Root Orthogonal Matrix
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