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Questions for repetition: diagonalization by orthogonal matrices

1. Describe the steps leading to a full definition of the set of complex numbers (imaginary unit, complex numbers, operations with them, absolute value, conjugate number).

2. What are matrix analogs of a) real numbers, b) conjugation and c) an absolute value of a complex number (without proof)?

3. Write out and prove the properties of the scalar product in $latex C^n$.

4. Assuming the Euler formula known, derive the polar form of a complex number. What can you say about the angle $latex \theta$ in the polar form?

5. Prove that a quadratic form of a matrix is homogeneous of degree 2.

6. Prove that a quadratic form takes values in the set of numbers.

7. Define positive and non-negative matrices. State Sylvester's criterion.

8. Illustrate the Sylvester criterion geometrically.

9. Show that $latex A^TA$ is symmetric and non-negative.

10. Give the definition of a basis and derive the formula for the coefficients $latex \xi$ in the decomposition $latex x=\sum\xi_iu_i$ for an arbitrary vector $latex x$.

11. How are the decomposition coefficients transformed when one passes from one basis to another?

12. Give the full picture behind the similarity definition.

13. Prove that for an orthogonal matrix, a) the inverse and transpose are the same, b) the transpose and inverse are orthogonal.

14. An orthogonal matrix preserves scalar products, norms and angles.

15. If you put elements of an orthonormal basis side by side, the resulting matrix will be orthogonal.

16. The transition matrix from one orthonormal basis to another is orthogonal.

17. Show that the product of two diagonal matrices is diagonal.

18. Define eigenvalues and eigenvectors. Why are we interested in them?

19. What is the link between eigenvalues and a characteristic equation of a matrix?

20. Prove that the characteristic equation is a polynomial of order $latex n$, if $latex A$ is of size $latex n\times n$.

21. Prove that in $latex C^n$ any matrix has at least one eigenvector.

22. A symmetric matrix in $latex C^n$ has only real eigenvalues.

23. If $latex A$ is symmetric, then it has at least one real eigenvector.

24. A symmetric matrix has at least one eigenvector in any nontrivial invariant subspace.

25. What is the relationship between the spectra of $latex A_C$ and $latex A_R$?

26. A matrix diagonalizable by an orthogonal matrix must be symmetric.

27. For a symmetric matrix, an orthogonal complement of an invariant subspace is invariant.

28. Main theorem. A symmetric matrix is diagonalizable by an orthogonal matrix.

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