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Basic Maths / Square Root of Matrices
« on: May 31, 2013, 11:27:58 AM »
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 are , 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 can be diagonalized as , where and . D has principal square root , giving the square root .
Ref: Wikipedia
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 can be diagonalized as , where and . D has principal square root , giving the square root .
Ref: Wikipedia