![]() Linear Algebra: Eigenvalues, Eigenvectors, Cayley-Hamilton Theorem Linear Algebra - Eigenvalues, Eigenvectors and Cayley Hamilton Theorem - Outline of Contents:
Here's a quick outline of topics to be covered in this tutorial: RankIf all the minors of the matrix A of order r+1 are zero and at least one minor of order r≠0 then matrix A is said to be of rank ‘r’Echelon form of MatrixA matrix ’A’ is said to be in echelon form if,1. All the zero rows of ‘A’ followed by non-zero row 2. The number of zero’s before the first non-zero element in a non-zero row is less than the number of such zeros in the next row 3. The first non-zero element in a non-zero row is ‘1’ Elementary Transformations or Elementary operations of a matrix -and what they result in:The following three operations applied on the rows (columns) of a matrix are called elementary row (column) transformations. We'll take a look at what happens after making these elementary transformations.1. Interchange of any two rows(columns) 2. Multiplying all elements of a row(column) of a matrix by a non-zero scalar 3. Adding to the elements of a row (column) ,the corresponding elements of any other row(column) multiplied by any scalar k. A major focus of this tutorial will be to introduce Eigenvalues and Eigenvectors
We'll also introduce important theorems such as :Rank of the matrix ‘A’ is the number of non-zero rows in its echelon formThe product of the Eigen values of a square matrix A is |A| Cayley Hamilton Theorem: Every square matrix of order n satisfies its own characteristic equation ... and so on Complete Tutorial : Understanding Eigenvalues and Eigen-vectors more intuitively: And from an answer I gave on Quora. The real world application of these concepts is quite fascinating - in the domain of Social, Economic or Internet based Networks, and connected agents in a system. Networks over here doesn't necessarily mean computer or communication networks, just the way in which people are linked. Think of a "network" as an NxN matrix, which has information about how N people are connected to each other. The Adjacency Matrix is an NxN matrix, let's say it looks something like this. People who aren't connected to each other have A[i][j] = 0, people with weak relationships have A[i][j] = 0.1, people with medium ties have A[i][j]=0.4, and people with strong links have A[i][j] = 0.6 1---2----3----4 1---0.0 0.1 0.4 0.6 2---0.1 0.0 0.4 0.0 3---0.4 0.4 0.0 0.1 4---0.6 0.0 0.1 0.0 If we look at the chart above, 1 and 2 are weakly connected, 1 and 3 have medium ties, 1 and 4 have strong ties. This is just a quick example to give you a quick idea. These matrices may not always be symmetric either. People with "higher eigenvector centrality" are people who are better connected to each other. This takes into account, not just how many people the person knows, but also whom the person knows. If a person is connected to a few well connected people, that might fetch him more centrality than a person connected to a lot of people who are not so well connected. This is a kind of a recursive function, similar to the page/priority rank used by search engines to prioritize websites and pages. Social and Economic Networks Page on Mit The above is a case study, based on an experiment in Karnataka about how Micro-finance diffuses. Micro-finance involves small amount of capital; is used to help people set up small businesses; and it can potentially pull a lot of people out of poverty. It is found, that when Micro-finance is injected by introducing it to people who have higher eigenvector centrality, it spreads much more quickly. In the rural setting where this experiment was conducted, this eigenvector centrality will be higher for someone like the priest of a temple or a community head, or the person who leads the local governance body. Similarly, if a product or service needs to be advertised, if it is introduced to "better connected" people with higher eigenvector-centrality, it is likely to see a better and faster adoption rate. You might like to take a look at some of our other Linear Algebra tutorials :
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