# Lecture 27 Positive Definite and its Graphical analysis

1. How to check whether it is positive definite.

# Lecture 24: Markov and Fourier.

1. Markov matrice:

# Lecture 21. Eigen Value and Eigen Vector.

1. What a matrix do? It multiplies to a vector, and give out another vector. It is like a function.
2. The eigenvectors are the vectors that went through the matrix but still remain the same direction. A*x parallel to x.
3. Ax = λx, λ can be zero or negative. Ax = 0: Nullspace. X is the eigenvector, λ is the eigenvalue.
4. If A is singular, then λ=0 is the eigenvalue.
5. Ax=λx, Elimination doesn’t work here, there are two unknowns, x and λ.
6. From the projection perspective, if a vector x is projected into some plane, and the direction of that vector…

# Lecture 18. Determinant.

1. Det(A), |A|, determinant has signs +,-.
2. Invertibility. Det(A) > 0 , invertible.
3. Sometimes, determinant feels like the area.
4. First Property: Det(I) = 1
5. Second Property: Row exchanging reverse the sign of the determinant.
6. A permutation matrix is a row exchanging of I. So the determinant of Permutation matrices is +-1.
7. How to compute in 2D

# Lecture 16 Projection Matrices and Least Square

1. b in column space: project something already in the column space.

# Lecture 12. An application in Physics: Represent Graph with Matrix.

1. 4 Nodes, 5 Edges.

# Lecture 9. Independence, Basis, and Dimension with Nullspace.

1. A with m rows, n columns, m <<n. Means we have more unknown variables than data. There are non-zero solutions to Ax = 0. Because we have more free columns. And free variables.
2. Independent vector: c1 * x1 + c2 * x3 + …

# Lecture 7 Ax= 0, Nullspace

1. Elimination of A. Rank = Number of Pivot in the Matrix. 