This captured the low-frequency part of the signal, and filtered out the high-frequency noise. That’s just what we wanted! This is called a low-pass filter.
We can zoom in on a few points to see the effect of the filter.
Now you try
See if you can make a figure which will just find the noisyness…
A different filter
What if we subtract the values at x_{k+1} and x_{k-1} instead of adding them?
This is a high-pass filter. It captures the high-frequency noise, but filters out the low-frequency signal.
Inverse Matrices
Definition of inverse matrix
Let A be a square matrix.
Inverse for A is a square matrix B of the same size as A
such that A B=I=B A.
If such a B exists, then the matrix A is said to be invertible.
Also called “singular” (non-invertible), or “nonsingular” (invertible)
Conditions for invertibility
Conditions for Invertibility The following are equivalent conditions on the square n \times n matrix A :
The matrix A is invertible.
There is a square matrix B such that B A=I.
The linear system A \mathbf{x}=\mathbf{b} has a unique solution for every right-hand-side vector \mathbf{b}.
The linear system A \mathbf{x}=\mathbf{b} has a unique solution for some right-hand-side vector \mathbf{b}.
The linear system A \mathbf{x}=0 has only the trivial solution.
\operatorname{rank} A=n.
The reduced row echelon form of A is I_{n}.
The matrix A is a product of elementary matrices.
There is a square matrix B such that A B=I.
Elementary matrices
Each of the operations in row reduction can be represented by a matrix E.
Remember: - E_{i j} : The elementary operation of switching the ith and jth rows of the matrix. - E_{i}(c) : The elementary operation of multiplying the ith row by the nonzero constant c. - E_{i j}(d) : The elementary operation of adding d times the jth row to the ith row.
Find an elementary matrix of size n is by performing the corresponding elementary row operation on the identity matrix I_{n}.
Example: Find the elementary matrix for E_{13}(-4)
Add -4 times the 3rd row of I_{3} to its first row…
B=E_{k} E_{k-1} \cdots E_{1} is the product of the various elementary matrices we used.
Inverse Algorithm
Given an n \times n matrix A, to compute A^{-1} :
Form the superaugmented matrix \widetilde{A}=\left[A \mid I_{n}\right].
Reduce the first n columns of \tilde{A} to reduced row echelon form by performing elementary operations on the matrix \widetilde{A} resulting in the matrix [R \mid B].
If R=I_{n} then set A^{-1}=B; otherwise, A is singular and A^{-1} does not exist.
2x2 matrices
Suppose we have the 2x2 matrix
A=\left[\begin{array}{ll}
a & b \\
c & d
\end{array}\right]
Do row reduction on the superaugmented matrix:
\left[\begin{array}{ll|ll}
a & b & 1 & 0 \\
c & d & 0 & 1
\end{array}\right]
A^{-1}=\frac{1}{D}\left[\begin{array}{rr}
d & -b \\
-c & a
\end{array}\right]
PageRank revisited
Back to PageRank
Do page ranking as in the first week:
for page j let n_{j} be its total number of outgoing links on that page.
Then the score for vertex i is the sum of the scores of all vertices j that link to i, divided by the total number of outgoing links on page j. . . .
\begin{equation*}
x_{i}=\sum_{x_{j} \in L_{i}} \frac{x_{j}}{n_{j}}
\end{equation*}
We can introduce a correction vector, equivalent to adding links from the dangling node to every other node, or to all connecting nodes (via some path).
