Questions for repetition: diagonal representation
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Let
be the eigenvalues of a symmetric matrix
Prove that a) it is positive if and only if its eigenvalues are positive and b) it is non-negative if and only if its eigenvalues are non-negative. Reproduce my proof, which gives only sufficiency (
implies positivity of
and
implies non-negativity of
). For the necessity part, plug eigenvectors of
in
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Use the Cauchy-Hadamard theorem to relate the radius of convergence of the power series used to define a function of a matrix
to the radius of convergence of the function
of a numerical argument.
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What is the matrix solution of the initial value problem
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How does the knowledge of the diagonal representation of
simplify finding
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Describe how the knowledge of the diagonal representation of
allows one to split the system
into a collection of one-dimensional equations. How does this lead to the solution of the initial value problem in Exercise 3?
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Define a square root of a non-negative symmetric matrix and relate it to the definition of the same using the power series. What are the properties of the square root?
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Show that the variance matrix
of an arbitrary random vector
(with real random components) is symmetric and non-negative.
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When the matrix
is positive, what are the properties of
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Find the variance of
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How does the previous result lead to the Aitken estimator?
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Define the absolute value of
and show that this definition is correct.
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If
is diagonalized, what is the expression of its determinant in terms of its eigenvalues?
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(This is strictly about ideas) Describe the elements of the polar form
of a complex number. How do the definitions of
and
help you define their analogs for matrices?
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Derive the polar form for a square matrix.
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More on similarity between complex numbers and matrices. For a square matrix with possibly complex entries, define the real part
and the imaginary part
Here
is the adjoint of
Show that both
and
are symmetric and that
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Using population characteristics, describe the idea of Principal Component Analysis.
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Show how this idea is realized in the sampling context.
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This is a research problem to support Exercise 17. How do you change places of rows in a matrix? We want to find an orthogonal matrix
such that premultiplication of
by
yields a matrix
where
has the same elements as
except that some rows have changed their places. a) Consider a matrix
of size
and let
be the transformed matrix (with the first row of
as the second row of
and vice versa). Find
from the equation
b) Do the same for a matrix
of size
c) Generalize to the case of an
matrix
first considering matrices
that change places of only two rows. Let's call such a matrix an elementary matrix. Note that it is orthogonal. d) The matrix that changes any number of rows is a product of elementary ones. It is orthogonal as a product of orthogonal matrices.
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