Raising the bar

Questions for repetition: diagonal representation

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

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

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

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

5. Describe how the knowledge of the diagonal representation of $latex A$ allows one to split the system $latex x^\prime(t)=Ax(t)$ 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 $latex \Omega $ of an arbitrary random vector $latex e$ (with real random components) is symmetric and non-negative.

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

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

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

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

12. If $latex A$ 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 $latex c=\rho e^{i\theta}$ of a complex number. How do the definitions of $latex \rho$ and $latex e^{i\theta }$ 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 $latex \text{Re}(A)=(A+A^\prime)/2$ and the imaginary part $latex \text{Im}(A)=(A-A^\prime)/(2i).$ Here $latex A^\prime$ is the adjoint of $latex A.$ Show that both $latex \text{Re}(A)$ and $latex \text{Im}(A)$ are symmetric and that $latex A=\text{Re}(A)+i\text{Im}(A).$

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