Linear Algebra

Basic Linear Algebra

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

[math]AB = BA[/math] (it commutes)
[math]AB \neq BA[/math] (does not commute)

In general, matrix multiplication is not commutative.

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

(Solution 8.3)

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

(Solution 8.5)

8-7. Prove that [math](A^{-1})^{-1} = A[/math].

(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 [math]M = L \cdot U[/math] has a zero in its diagonal.

(Solution 8.11)

Eigenvalues and Eigenvectors

8-13. Prove that the eigenvalues of A and [math]A^T[/math] are identical.

(Solution 8.13)

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

(Solution 8.15)

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

(Solution 8.17)

Implementation Projects

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

Speed: How does the run time compare, for both dense and sparse coefficient matrices?
Accuracy: What are the size of the numerical residuals [math]C X - Y[/math], particularly as the condition number of the matrix increases.
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)