symmetric matrix multiplication

[statfan]

I have a symmetric matrix B (17x17), and a (17x17) square matrix A.  If do the following matrix multiplication I SHOULD get a symmetric matrix, however i don't.  The computation required is:

C = t(A)%*%B%*%A

here are some checks for symmetry
> (max(abs(B - t(B))))
[1] 0
> C = t(A)%*%B%*%A
> (max(abs(C - t(C))))
[1] 3.552714e-15

Any help on the matter would be very much appreciated.

[Daniel Nordlund-4]

> -----Original Message-----
> From: [hidden email] [mailto:[hidden email]]
> On Behalf Of statfan
> Sent: Saturday, October 22, 2011 10:45 PM
> To: [hidden email]
> Subject: [R] symmetric matrix multiplication
>
> I have a symmetric matrix B (17x17), and a (17x17) square matrix A.  If do
> the following matrix multiplication I SHOULD get a symmetric matrix,
> however
> i don't.  The computation required is:
>
> C = t(A)%*%B%*%A
>
> here are some checks for symmetry
> > (max(abs(B - t(B))))
> [1] 0
> > C = t(A)%*%B%*%A
> > (max(abs(C - t(C))))
> [1] 3.552714e-15
>
> Any help on the matter would be very much appreciated.
>
>

Welcome to the world of floating-point calculation on finite precision computers.  You need to read R FAQ 7.31.  Your maximum difference is for all intents and purposes equal to zero.  

Hope this is helpful,

Dan

Daniel Nordlund
Bothell, WA USA

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[ted.harding-3]

On 23-Oct-11 07:00:07, Daniel Nordlund wrote:

>> -----Original Message-----
>> From: [hidden email]
>> [mailto:[hidden email]]
>> On Behalf Of statfan
>> Sent: Saturday, October 22, 2011 10:45 PM
>> To: [hidden email]
>> Subject: [R] symmetric matrix multiplication
>>
>> I have a symmetric matrix B (17x17), and a (17x17) square matrix A.
>> If do
>> the following matrix multiplication I SHOULD get a symmetric matrix,
>> however
>> i don't.  The computation required is:
>>
>> C = t(A)%*%B%*%A
>>
>> here are some checks for symmetry
>> > (max(abs(B - t(B))))
>> [1] 0
>> > C = t(A)%*%B%*%A
>> > (max(abs(C - t(C))))
>> [1] 3.552714e-15
>>
>> Any help on the matter would be very much appreciated.
>
> Welcome to the world of floating-point calculation on
> finite precision computers. You need to read R FAQ 7.31.
> Your maximum difference is for all intents and purposes
> equal to zero.
>
> Hope this is helpful,
> Dan
>
> Daniel Nordlund
> Bothell, WA USA

In addition to Dan's comment, let me point out that you
can convert your very nearly symmetric matrix C to an
exactly (even by R's finite-precision standards) symmetric
matrix by using (C + t(C))/2. The result will differ from
the original matrix C by similar "for all intents and purposes
zero" amounts. Here is an example, using 4x4 matrices:

##[1]: The symmetric matrix B:
B <- matrix( c(
  1.1, 1.2, 1.3, 1.4,
  1.2, 2.2, 2.3, 2.4,
  1.3, 2.3, 3.3, 3.4,
  1.4, 2.4, 3.4, 4.4), byrow=TRUE, nrow=4 )

##[2]: The non-symmetric matrix B:
A <- matrix( c(
  1.1, 1.2, 1.3, 1.4,
  2.1, 2.2, 2.3, 2.4,
  3.1, 3.2, 3.3, 3.4,
  4.1, 4.2, 4.3, 4.4), byrow=TRUE, nrow=4 )

##[3]: An allegedly symmetric matrix C1 (constructed
##     like your C):
C1 <- t(A)%*%B%*%A

##[4]: But it isn't exactly symmetric:
max(abs(C1 - t(C1)))
# [1] 5.684342e-14

##[5]: So construct an exactly symmetric version:
C2 <- (C1 + t(C1))/2

##[6]: Check that it is exactly symmetric:
max(abs(C2 - t(C2)))
# [1] 0

##[7]: And check how close it is to the original C1:
max(abs(C2 - C1))
# 1] 5.684342e-14

Hoping this helps!
Ted.

--------------------------------------------------------------------
E-Mail: (Ted Harding) <[hidden email]>
Fax-to-email: [phone]
Date: 23-Oct-11                                       Time: 08:43:18
------------------------------ XFMail ------------------------------

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and provide commented, minimal, self-contained, reproducible code.

[statfan]

Thank you Dan and Ted for these helpful comments. I will implement this simple force symmetry code you suggested and make sure I familiarize with this floating-point calculation problem so I can recognize such issues in the future.