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Unformatted text preview: ogonal if its column vectors b1 , b2 , . . . , bn satisfy the relations
b1 · b1 = 1, b1 · b2 = 0, · · · ,
b2 · b2 = 1, · · · , b1 · bn = 0,
b2 · bn = 0,
bn · bn = 1. Using this latter criterion, we can easily check that, for example, the matrices 1/3
cos θ − sin θ
−2/3 −1/3 2/3
2/3 −2/3 1/3
are orthogonal. Note that since
BT B = I ⇒ (det B )2 = (det B T )(det B ) = det(B T B ) = 1, the determinant of an orthogonal matrix is always ±1.
Recall that the eigenvalues of an n × n matrix A are the roots of the polynomial equation
det(A − λI ) = 0.
For each eigenvalue λi , the corresponding eigenvectors are the nonzero solutions
x to the linear system
(A − λI )x = 0.
For a general n × n matrix with real entries, the problem of ﬁnding eigenvalues
and eigenvectors can be complicated, because eigenvalues need not be real (but
can occur in complex conjugate pairs) and in the “repeated root” case, there
may not be enough eigenvectors to construct a b...
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- Winter '14