Term
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Definition
Dimension: m edges x n nodes
A(i,j)=
−1, if edge i leaves node j
+1 if edge i enters node j
0 otherwise |
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| Meaning of nullspace of incidence matrix |
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Definition
| dim(Nul(A)) is number of connected subgraphs |
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| Meaning of left nullspace of incidence matrix |
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Definition
left nullspace = null(A^T)
dim(Nul(A^T)) is # of independent loops |
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| Orthogonal basis definition |
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Definition
| If the vectors are orthogonal to each other. |
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| Orthogonal projection of vector x ONTO vector y |
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Definition
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Term
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Definition
Used to project x into y by using Px
Find using: [image] |
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| Orthogonal projection of x onto W |
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Definition
Determined by xHat
[image]
Once xˆ is determined, x⊥ (error term) = x − xˆ. |
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| Projection matrix P for orthogonal projectino onto W in R^n (or other basis) |
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Definition
Use columns as matrix.
Each column is following the formula:
[image]
where x is in the non W basis (1,0,0) (0,1,0), (0,0,1) for the std basis R^3 |
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Definition
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| Closest point to x in span(v1,v2,..) |
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Definition
Follow this formula. Resulting vector/pt is closest)
[image] |
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| Find least squares solution to Ax = b (and define meaning of soln) |
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Definition
Solve [image]
[image] is minimal |
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| Projection matrix for proj onto Col(A) |
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Definition
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Definition
Solve for beta:
[image]
where:
[image]
[image]
[image]
Line is [image] |
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Definition
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| Given a basis a1, . . . , an, produce a orthogonal basis b1, . . . , bn and an orthonormal basis q1, . . . , qn. |
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Definition
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| The columns of Q are orthonormal means ___ |
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Definition
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Definition
| square matrix with orthonormal columns |
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| Gram Schmidt A = QR decomp |
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Definition
1) Gram-Schmidt on columns of A to get columns of Q
2) R = Q^T*A
Q is orthonormal, R is upper triangular |
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| Least square solution using QR decomp (Ax = b) |
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Definition
Find QR decomp.
Rx = Q^T b, x will be best sol'n |
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Definition
Used for relating coordinate vectors of different basis.
[image] |
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Definition
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| Determinant of matrix (non std way) |
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Definition
1) Get into upper triangular matrix
2) multiply diagonal |
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| Solve for eigenvalues (λ) of matrix A |
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Definition
| det(A - λ * Identity) = 0 |
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Definition
| Ax = λx (x is a eigenvector, λ is eigen value) |
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| Product of eigenvalues is |
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Definition
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Definition
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Definition
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