Oct 18

Questions for repetition

Questions for repetition: diagonal representation

  1. Let \lambda_i be the eigenvalues of a symmetric matrix A. 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 (\min_i\lambda_i>0 implies positivity of A and \min_i\lambda_i\geq 0 implies non-negativity of A). For the necessity part, plug eigenvectors of A in (Ax)\cdot x.

  2. Use the Cauchy-Hadamard theorem to relate the radius of convergence of the power series used to define a function of a matrix f(A) to the radius of convergence of the function f(t) of a numerical argument.

  3. What is the matrix solution of the initial value problem x^\prime(t)=Ax(t), x(t_0)=x_0?

  4. How does the knowledge of the diagonal representation of A simplify finding f(A)?

  5. Describe how the knowledge of the diagonal representation of A allows one to split the system x^\prime(t)=Ax(t) into a collection of one-dimensional equations. How does this lead to the solution of the initial value problem in Exercise 3?

  6. 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?

  7. Show that the variance matrix \Omega of an arbitrary random vector e (with real random components) is symmetric and non-negative.

  8. When the matrix \Omega is positive, what are the properties of \Omega^{-1/2}?

  9. Find the variance of \Omega ^{-1/2}e.

  10. How does the previous result lead to the Aitken estimator?

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

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

  13. (This is strictly about ideas) Describe the elements of the polar form c=\rho e^{i\theta} of a complex number. How do the definitions of \rho and e^{i\theta } help you define their analogs for matrices?

  14. Derive the polar form for a square matrix.

  15. More on similarity between complex numbers and matrices. For a square matrix with possibly complex entries, define the real part \text{Re}(A)=(A+A^\prime)/2 and the imaginary part \text{Im}(A)=(A-A^\prime)/(2i). Here A^\prime is the adjoint of A. Show that both \text{Re}(A) and \text{Im}(A) are symmetric and that A=\text{Re}(A)+i\text{Im}(A).

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

  17. Show how this idea is realized in the sampling context.

  18. 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 P such that premultiplication of W by P yields a matrix W_1 where W_1 has the same elements as W except that some rows have changed their places. a) Consider a matrix W of size 2\times 2 and let W_1 be the transformed matrix (with the first row of W as the second row of W_1 and vice versa). Find P from the equation PW=W_1. b) Do the same for a matrix W of size 3\times 3. c) Generalize to the case of an n\times n matrix W, first considering matrices P 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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