Questions for repetition: diagonal representation

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 nonnegative if and only if its eigenvalues are nonnegative. Reproduce my proof, which gives only sufficiency ( implies positivity of and implies nonnegativity of ). For the necessity part, plug eigenvectors of in

Use the CauchyHadamard 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.

What is the matrix solution of the initial value problem

How does the knowledge of the diagonal representation of simplify finding

Describe how the knowledge of the diagonal representation of allows one to split the system into a collection of onedimensional equations. How does this lead to the solution of the initial value problem in Exercise 3?

Define a square root of a nonnegative symmetric matrix and relate it to the definition of the same using the power series. What are the properties of the square root?

Show that the variance matrix of an arbitrary random vector (with real random components) is symmetric and nonnegative.

When the matrix is positive, what are the properties of

Find the variance of

How does the previous result lead to the Aitken estimator?

Define the absolute value of and show that this definition is correct.

If is diagonalized, what is the expression of its determinant in terms of its eigenvalues?

(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?

Derive the polar form for a square matrix.

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

Using population characteristics, describe the idea of Principal Component Analysis.

Show how this idea is realized in the sampling context.

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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