Daffodil International University
Educational => Mathematics => Basic Maths => Topic started by: msu_math on May 31, 2013, 11:27:58 AM
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Like numbers, definition of square root can be extended for matrices. A matrix B is said to be a square root of A if B2=A. For example square roots of the matrix (http://upload.wikimedia.org/math/3/c/f/3cf9303cb7a004cbfa29aab00f6f3209.png) are (http://upload.wikimedia.org/math/b/f/9/bf9cd892194b0686fe62f4786c5ad9d7.png), (http://upload.wikimedia.org/math/5/8/8/58899d5862f9fb03868fea60f7887a74.png) and their additive inverses.
There are a number of methods for calculating square roots of matrices. One of the frequently used method is diagonalization method. An n×n matrix A is diagonalizable if there is a matrix V and a diagonal matrix D such that A = VD-1V . This happens if and only if A has n eigenvectors which constitute a basis for Cn. In this case, V can be chosen to be the matrix with the n eigenvectors as columns, and a square root of A is VSV-1, where S is any square root of D. Indeed, (VD1/2V-1)2 = VD1/2(V-1V)D1/2V-1 = VDV-1 = A .
As in the example, the matrix (http://upload.wikimedia.org/math/c/0/0/c004f0c054372d3d15c4d7f54489c868.png) can be diagonalized as (http://upload.wikimedia.org/math/4/d/1/4d15206bfebe1bfcc65a1fc7bfd1a678.png) , where (http://upload.wikimedia.org/math/1/1/e/11e70684b9d9e37e3875debcd4c020ec.png) and (http://upload.wikimedia.org/math/2/3/2/232c7774435b47794cc8832fa467410a.png). D has principal square root (http://upload.wikimedia.org/math/9/a/0/9a0fb539de78f3ce00abefd1a598e646.png) , giving the square root (http://upload.wikimedia.org/math/e/b/2/eb2ab2b6d8d2d5ae38c9ac98684cc34f.png).
Ref: Wikipedia
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Thank you sir.it so much important for us........
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bah!
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Nice to know...........
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Good post.