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Unformatted text preview: metric matrices Before proceeding further, we need to review and extend some notions from
vectors and matrices (linear algebra), which the student should have studied
in an earlier course. In particular, we will need the amazing fact that the
eigenvalueeigenvector problem for an n × n matrix A simpliﬁes considerably
when the matrix is symmetric.
An n × n matrix A is said to be symmetric if it is equal to its transpose AT .
Examples of symmetric matrices include 3−λ
6
5
abc
13
6
b d e .
1−λ
0
,
and
31
5
0
8−λ
cef
Alternatively, we could say that an n × n matrix A is symmetric if and only if
x · (Ay) = (Ax) · y. (2.1) for every choice of nvectors x and y. Indeed, since x · y = xT y, equation (2.1)
can be rewritten in the form
xT Ay = (Ax)T y = xT AT y,
which holds for all x and y if and only if A = AT .
On the other hand, an n × n real matrix B is orthogonal if its transpose is
equal to its inverse, B T = B −1 . Alternatively, an n × n matrix
B = (b1 b2 · · · bn ) 29 is orth...
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 Winter '14
 Equations

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