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**Unformatted text preview: **an be
uniquely factored as A = RU, where R is hermitian positive deﬁnite
and U is unitary. This is the matrix analog of the polar form of a
complex number z = reiθ , r > 0, because 1 × 1 hermitian positive
deﬁnite matrices are positive real numbers, and 1 × 1 unitary matrices
are points on the unit circle. Hint: First explain why R = (AA∗ )1/2 . IG
R 7.6.8. Explain why trying to produce better approximations to the solution
of the Dirichlet problem in Example 7.6.2 by using ﬁner meshes with
more grid points results in an increasingly ill-conditioned linear system
Lu = g. Y
P 78 7.6.9. For a given function f the equation ∇2 u = f is called Poisson’s
equation. Consider Poisson’s equation on a square in two dimensions
with Dirichlet boundary conditions. That is, O
C ∂2u ∂2u
+ 2 = f (x, y )
∂x2
∂y 78 with u(x, y ) = g (x, y ) on the boundary. Sim´on Denis Poisson (1781–1840) was a proliﬁc French scientist who was originally encouraged
e
to study medicine but was seduced by mathematics. While he was still a teenager, his work
attracted the attention of the reigning scientiﬁc elite of France such as Legendre, Laplace, and
Lagrange. The latter two were originally his teachers (Lagrange was his thesis director) at the
´
Ecole Polytechnique, but they eventually became his friends and collaborators. It is estimated
that Poisson published about 400 scientiﬁc articles, and his 1811 book Trait´ de m´canique was
e
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the standard reference for mechanics for many years. Poisson began his career as an astronomer,
but he is primarily remembered for his impact on applied areas such as mechanics, probability,
electricity and magnetism, and Fourier series. This seems ironic because he held the chair of
“pure mathematics” in the Facult´ des Sciences. The next time you ﬁnd yourself on the streets
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of Paris, take a stroll on the Rue Denis Poisson, or you can check out Poisson’s plaque, along
with those of Lagrange, Laplace, and Legendre, on the ﬁrst stage of the Eiﬀel Tower. Copyright c 2000 SIAM Buy...

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