Lecture 21. Eigen Value and Eigen Vector.
- What a matrix do? It multiplies to a vector, and give out another vector. It is like a function.
- The eigenvectors are the vectors that went through the matrix but still remain the same direction. A*x parallel to x.
- Ax = λx, λ can be zero or negative. Ax = 0: Nullspace. X is the eigenvector, λ is the eigenvalue.
- If A is singular, then λ=0 is the eigenvalue.
- Ax=λx, Elimination doesn’t work here, there are two unknowns, x and λ.
- From the projection perspective, if a vector x is projected into some plane, and the direction of that vector is not changed, the original x must be already in the plane. Vectors perpendicular to the plane will be projected as a point. So it is an eigenvalue and eigenvector as well.
7. Fact: the sum of eigenvalues are equal to the values on diagonal.
8. How to solve Ax=λx. Move to one side, take x out. Consider the λ first.
9. Example: Use the ad-bc = det(A). λ = 2 or λ = 4. Then find the eigenvector. Det(A-λI) = 0 is important.
10. Find the eigenvector by using (A-λI)x = 0. Find the nullspace of A-λI.
11. Add n*I to the matrix A, eigenvalue will change, but eigenvectors are not changing.
12. This one is not standing. Due to the fact that x may not be the eigenvector for B. Eigenvalue is not linear, they don’t multiply.
13. Rotation matrix. There is a conflict of trace and determinant. However, it is still solvable with a complex number. It reminds me of the complex number when doing camera transformation matrices.
14.Complex numbers: Conjugate with each other. Symmetric, wouldn’t happen. Anti-Symmetric: Q^T = -Q
15. Situation that no good. Only one eigenvector exist. Degenerated matrix.
Lecture 22. Diagonalization
- Ax=λx, S’s columns are n independent(S is invertible) eigenvectors of A.
S is corresponding to the x in Ax=λx. Λ is the diagonal eigenvalue matrix.
2. A is Diagonalized to (S) and (Λ), S are eigenvectors, Λ is eigenvalues.
3. Multiplication of A. Lambda is squared. x remains the same. k Powers of A, the eigenvector is not changing, the eigenvalue is k powered.
This diagonalization only happen when S^-1 make sense, i.e S is invertible, there are independent eigenvectors inside it.
4. The power of the matrix. A new view of the power of the matrix: Only change eigenvalue, eigenvector no changing.
Stable matrix: because the eigenvalues are less than one.
5. A sure to have n independent eigenvectors in S when all the eigenvalues are different. No repeated eigenvalues. Repeated eigenvalues may or may not be independent. E.g A is the Identity matrix.
6.Algebric multiplicity(From the root of the polynomial) is different from geometric multiplication(A-λI = 0). Two same eigenvalue result case. There are no two independent eigenvectors, only one here.
7. Solve this first-order difference equation(Only break down one level)
8. Fibonacci Example
9. All in all
First-order system u k+1= A * u_k
Find the eigenvector and eigenvalue of A.
Rewrite the u0 into a combination of EigenVectors * c
Lecture 23. Differential Equations. Solve Linear system.
0. It is important to know the eigenvalue of a matrix. It will decide whether this differential equation is stable or not.
- The solution to the constant-coefficient linear equation is exponential. What is the exponent, what is the multiplier?
2. Compute the eigenvalue and eigentvectors first.
8. Convert the derivative to u = A u, v = B v.
10. This series always converge, as the terms go on.
11. Why (I — At)^-1? From the above image, geometric series. 1/(1-x). Find the inverse of a matrix?
5th order: 5x5 matrix.