Class 29
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Hermitian Matrices and Diagonalization
(Sections 6.4)
Beware: The book treats only the special case that A is real and symmetric.
We are treating the more general case of complex matrices that are Hermitian.
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Theorems:
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All the roots of A are real.
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A is always diagonalizable
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A can always be diagonalized using a unitary matrix.
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Problems:
Diagonalize the following matrices, where i is the sqrt(-1).
(Find the unitary (orthogonal in the real case)
matrix that does the diagonalization):
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A= [ 0 i -i ]
[ -i 1 2 ]
[ i 2 1 ]
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B= [ 1 0 2 ]
[ 0 -1 -2 ]
[ 2 -2 0 ]
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Find all orthogonal matrices that diagonalize
C= [ 9 12 ]
[ 12 16 ]
For your interest only!
Concerning the ratios of the volumes discussed in class; if you want to be
convinced that the (small) ball in the center escapes from the box,
consider the following. Take n=6. Let x=(1 1 1 1 1 1) be the center of the
big ball in the first quadrant. ||x||=6. The radius of the ball is 1, since that
is the distance to the line through (1 0 0 0 0 0). Now take the point and use the
point on the ball y=(sqrt(6)-1) (x/sqrt(6)), i.e. y is in the direction of x and
has lenght sqrt(6)-1. Therefore the radius of the (small) ball in the center is
(sqrt(6)-1) > 2. But the point (sqrt(6)-1)(1 0 0 0 0 0) is on this ball and is
outside the box.