Linear-algebra-TDSM
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.
8-5.
Prove that if [math]AC = CA[/math] and [math]BC = CB[/math], then
[math]C(AB + BA) = (AB + BA)C[/math]
8-7.
Prove that [math](A^{-1})^{-1} = A[/math].
8-9.
Is the LU factorization of a matrix unique? Justify your answer.
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.
Eigenvalues and Eigenvectors
8-13.
Prove that the eigenvalues of A and [math]A^T[/math] are identical.
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]?
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?
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 \\(C X - Y\\), 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?
Interview Questions
8-21.
What is singular value decomposition? What is a singular value? And what is a singular vector?
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
8-25.
Identify mobile phone users from accelerometer data.
https://www.kaggle.com/c/accelerometer-biometric-competition
