This is our first example of a numerical technique which is based on iteration. Therefore, a random vector u may be written as:. Thus, if this sequence approaches the maximum eigenvector v 1. Now, we use the property of the norm that: The howto describes the numerical technique for finding this largest eigenvalue and its corresponding eigenvector. We assume that the largest eigenvalue is real and positive and that the matrix is diagonalizable. We begin with a random unit vector v 0.
There are certain vectors for which this method will not converge to the largest eigenvalue, however, the probability of choosing such a vector randomly is zero. If we halt due to Condition 2, then either the largest eigenvalue is complex or the two largest eigenvalues are very close in absolute value.
You will note that with every iteration, we divided the vector by its absolute value. While in theory, we could simply continue multiplying v 0 by higher and higher powers of M and look at the relative norms, because we are using the double format to store numbers, the growth in the magnitude will be exponential, and thus we will very quickly either overflow or underflow. These are shown in Figure 1. The sequence of approximations is shaded from blue to red. All of Quantum Mechanics is based on the notion of eigenvectors and eigenvalues. Observables are represented by hermitian operators Q, their determinate states are eigenvectors of Q, a measure of the observable can only yield an eigenvalue of the corresponding operator Q.
And so on and so forth. Of course Quantum Physics is not mathematically trivial: However, in the extremely fast growing field of Quantum Computing the algebra is mostly limited to finite-dimensional spaces and their operators.
Application of Eigenvalues and eigenvectors in Electrical Engineering | Linear Algebra
Finally, let me mention that Frank Wilczek, a winner of the Nobel Prize in Physics, has interestingly reminisced that as a student he found Quantum Mechanics easier than Classical Mechanics because of its nice axiomatization alluded to above.. For visual appeal, you should look into the area of pendulums. There is a good demonstration with swinging bottles, I recall, and this does depend on eigenvalues that are nearly equal. Do a Web search on "coupled pendulums". Principal Component Analysis is a way of identifying patterns in data, and expressing the data in such a way as to highlight their similarities and differences.
Applications of Eigenvalues and Eigenvectors
It is very difficult to visualize data in high dimensional space, but PCA can be used their to analyze data. From the data set covariance matrix is formed and then eigen values and eigen vectors of that covariance matrix are found. These eigne values and eigen vectors then can be compared to figure out the contribution of a particular feature in the data set. Thus PCA can be successfully applied to reduce dimension of the data.
Measurements of Social Influence. Another interesting application is rigid body rotation theory.
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Eamon Barkhordarian. More From Dr. Extension Lectures Connectedness and Disconnectedness in topological spaces.
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Related applications of eigenvalues and eigenvectors in electrical engineering
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