I have been reading the Vector iteration/power method for computing the Eigen values and eigen vectors and have got some questions. I shall be grateful if someone can help me.(adsbygoogle = window.adsbygoogle || []).push({});

Now, in power method we get the dominant eigen value and corresponding eigen vector.

This is followed by deflation method wherein we compute the reamining Eigen values and eigen vectors.

To compute the dominant eigen value and corresponding eigen vector we carry out the following steps:

i. Assign to the candidate matrix an arbitrary eigenvector with at least one element being nonzero.

ii. Compute a new eigenvector.

iii. Normalize the eigenvector, where the normalization scalar is taken for an initial eigenvalue.

iv. Multiply the original matrix by the normalized eigenvector to calculate a new eigenvector.

v. Normalize this eigenvector, where the normalization scalar is taken for a new eigenvalue.

vi. Repeat the entire process until the absolute relative error between successive eigenvalues satisfies an arbitrary tolerance (threshold) value.

My questions here is:

1) How can it be said that the eigen value is the dominant eigen value? I mean what is the physical interpretation of it being dominant?

Then in deflation emthod we do the following:

First, we use the Power Method to find the largest eigenvalue and eigenvector of matrix A.

a) multiply the largest eigenvector by its transpose and then by the largest eigenvalue. This produces the matrix Z* = c *X*(X)T

b) compute a new matrix A* = A - Z* = A - c *X*(X)T

c) Apply the Power Method to A* to compute its largest eigenvalue. This in turns should be the second largest eigenvalue of the initial matrix A.

Though mathematically easy to do/program these steps but what is it exactly are we doing in a),b) and c). IS it some kind of similarity transformation?

Please help

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# Vector iteration/Power method for eigen values

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