R help - Jan 2007 - Matrix question: obtaining the square root of a positive definite matrix?

I want to compute B=A^{1/2} such that B*B=A.

For example

 a=matrix(c(1,.2,.2,.2,1,.2,.2,.2,1),ncol=3)

so
> a
     [,1] [,2] [,3]
[1,]  1.0  0.2  0.2
[2,]  0.2  1.0  0.2
[3,]  0.2  0.2  1.0
> a%*%a
     [,1] [,2] [,3]
[1,] 1.08 0.44 0.44
[2,] 0.44 1.08 0.44
[3,] 0.44 0.44 1.08
> b=a%*%a
i have tried to use singular value decomposion
> c=svd(b)
> c$u%*%diag(sqrt(c$d))
           [,1]          [,2]       [,3]
[1,] -0.8082904  2.043868e-18  0.6531973
[2,] -0.8082904 -5.656854e-01 -0.3265986
[3,] -0.8082904  5.656854e-01 -0.3265986

this does not come close to the original a. Can anybody on this forum
enlight me on how to get a which is the square root of b?

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On Wed, 24 Jan 2007, gallon li wrote:
> I want to compute B=A^{1/2} such that B*B=A.
According to your subject line A is positive definite and hence 
symmetric?  The usual definition of a matrix square root involves a 
transpose, e.g. B'B = A.  There are many square roots: were you looking 
for a symmetric one?

For such an A,
> e <- eigen(A)
> V <- e$vectors
> V %*% diag(e$values) %*% t(V)
recovers A (up to rounding errors), and
> B <- V %*% diag(sqrt(e$values)) %*% t(V)
is such that B %*% B = A.  Even that is not unique, e.g. -B is an equally 
good answer.

> For example
(with A = b and B = a, it seems)
> a=matrix(c(1,.2,.2,.2,1,.2,.2,.2,1),ncol=3)
>
> so
>> a
>     [,1] [,2] [,3]
> [1,]  1.0  0.2  0.2
> [2,]  0.2  1.0  0.2
> [3,]  0.2  0.2  1.0
>> a%*%a
>     [,1] [,2] [,3]
> [1,] 1.08 0.44 0.44
> [2,] 0.44 1.08 0.44
> [3,] 0.44 0.44 1.08
>> b=a%*%a
>
> i have tried to use singular value decomposion
>
>> c=svd(b)
>
>> c$u%*%diag(sqrt(c$d))
>           [,1]          [,2]       [,3]
> [1,] -0.8082904  2.043868e-18  0.6531973
> [2,] -0.8082904 -5.656854e-01 -0.3265986
> [3,] -0.8082904  5.656854e-01 -0.3265986
>
> this does not come close to the original a. Can anybody on this forum
> enlight me on how to get a which is the square root of b?
>
> 	[[alternative HTML version deleted]]
>
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-- 
Brian D. Ripley,                  ripley at stats.ox.ac.uk
Professor of Applied Statistics,  http://www.stats.ox.ac.uk/~ripley/
University of Oxford,             Tel:  +44 1865 272861 (self)
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