# [R] Find A, given B where B=A'A

Charles C. Berry cberry at tajo.ucsd.edu
Wed Oct 31 22:33:44 CET 2007

```On Thu, 1 Nov 2007, Rolf Turner wrote:

>
> On 1/11/2007, at 9:13 AM, Michael Gormley wrote:
>
>> Given a matrix B, where B=A'A, how can I find A?
>> In other words, if I have a matrix B which I know is another matrix
>> A times
>> its transpose, can I find matrix A?
>
> You can't, because A is not unique.  You can easily find ***a***
> solution.
>
> E.g. A1 <- matrix(1:4,ncol=2)
>      B  <- t(A1)%*%A1
>      A2 <- msqrt(B)

Also, see

?chol

Chuck

>
> A2 != A1 (A2 is symmetric), yet t(A2)%*%A2 == B.
>
> The function msqrt() above is a simple-minded calculation of the
> square root of a positive semi-definite real matrix, the code of
> which I just cribbed from an old posting by Prof. Brian Ripley:
>
> msqrt <- function(X) {
> 	e <- eigen(X)
> 	V <- e\$vectors
> 	V%*%diag(sqrt(e\$values))%*%t(V)
> }
>
> The problem of finding ***all possible*** solutions A to A'A = B
> (for B symmetric positive semi-definite) is likely to be hard,
> but may have been solved by the linear algebraists.  I dunno.
>
> 		cheers,
>
> 			Rolf Turner
>
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