Linear-algebra-TDSM

Linear Algebra

Basic Linear Algebra

8-1. Give a pair of square matrices A and B such that:

1. $$AB = BA$$ (it commutes)
2. $$AB \neq BA$$ (does not commute)

In general, matrix multiplication is not commutative.

(Solution 8.1)

8-3. Prove that matrix multiplication is associative, i.e. that $$(AB)C = A(BC)$$ for compatible matrices A, B and C.

(Solution 8.3)

8-5. Prove that if $$AC = CA$$ and $$BC = CB$$ , then $$C(AB + BA) = (AB + BA)C$$

(Solution 8.5)

8-7. Prove that $$(A^{-1})^{-1} = A$$ .

(Solution 8.7)

8-9. Is the LU factorization of a matrix unique? Justify your answer.

(Solution 8.9)

8-11. Show that if M is a square matrix which is not invertible, then either L or U in the LU-decomposition $$M = L \cdot U$$ has a zero in its diagonal.

(Solution 8.11)

Eigenvalues and Eigenvectors

8-13. Prove that the eigenvalues of A and $$A^T$$ are identical.

(Solution 8.13)

8-15. Suppose that matrix A has an eigenvector v with eigenvalue $$\lambda$$ . Show that v is also an eigenvector for $$A^2$$ , and find the corresponding eigenvalue. How about for $$A^k$$ , for $$2 \leq k \leq n$$ ?

(Solution 8.15)

8-17. Show that the eigenvalues of $$MM^T$$ are the same as that of $$M^TM$$ . Are their eigenvectors also the same?

(Solution 8.17)

Implementation Projects

8-19. Implement Gaussian elimination for solving systems of equations, $$C \cdot X=Y$$ . Compare your implementation against a popular library routine for

1. Speed: How does the run time compare, for both dense and sparse coefficient matrices?
2. Accuracy: What are the size of the numerical residuals $$C X - Y$$ , particularly as the condition number of the matrix increases.
3. Stability: Does your program crash on a singular matrix? What about almost singular matrices, created by adding a little random noise to a singular matrix?

(Solution 8.19)

Interview Questions

8-21. What is singular value decomposition? What is a singular value? And what is a singular vector?

(Solution 8.21)

Kaggle Challenges

8-23. Tell what someone is looking at from analysis of their brain waves. https://www.kaggle.com/c/decoding-the-human-brain

(Solution 8.23)

8-25. Identify mobile phone users from accelerometer data. https://www.kaggle.com/c/accelerometer-biometric-competition

(Solution 8.25)