[R] identity matrix

[Thomas Lumley]

On Sun, 30 Oct 2005, Robert wrote:

> I found a very odd thing.
> A matrix multiplied by its inverse matrix should be an
> identity matrix.

Well, an invertible matrix multiplied by its inverse...

The matrix that you give (to two decimal places) is singular
> solve(a)
Error in solve.default(a) : system is computationally singular: reciprocal 
condition number = 3.25149e-20

Your matrix is slightly less singular, but the huge size of the elements 
in solve(a) shows that a%*%solve(a) will not be accurately computed.

There are limits to what you can do with only 15 digits accuracy.

 	-thomas


> But why the following thing happens?
> <a%*%solve(a) is not an identity matrix>
>> x%*%t(x)
>       [,1]   [,2]  [,3]   [,4]   [,5]
> [1,] 108.16  58.24 32.24  66.56 225.68
> [2,]  58.24  31.36 17.36  35.84 121.52
> [3,]  32.24  17.36  9.61  19.84  67.27
> [4,]  66.56  35.84 19.84  40.96 138.88
> [5,] 225.68 121.52 67.27 138.88 470.89
>> a=x%*%t(x)
>> solve(a)
>              [,1]          [,2]          [,3]
> [,4]          [,5]
> [1,] -1.372649e+14  1.078492e+14 -2.553747e+14
> 1.126842e+14  4.120191e+13
> [2,] -1.174558e+14  1.543860e+14  2.323143e+14
> -1.375074e+14  2.381809e+13
> [3,] -3.062129e+14  6.914056e+14 -1.656744e+15
> 7.007732e+14 -1.672741e+12
> [4,]  1.761208e+14 -3.724017e+14  7.773835e+14
> -2.156631e+14 -3.575351e+13
> [5,]  8.789836e+13 -8.046912e+13  6.984285e+13
> -5.502429e+13 -1.510937e+13
>> a%*%solve(a)
>           [,1]       [,2]      [,3]       [,4]
> [,5]
> [1,]  2.0410156  1.4433594 -4.169922  0.4003906
> 1.2170410
> [2,] -2.5146484  1.9208984 -1.84912w=1  1.1879883
> 1.1203613
> [3,] -1.5356445 -0.8549805 -1.008789  0.4116211
> 0.6218872
> [4,] -0.5898438 -0.2539063  3.592773  2.0351563
> 0.7849121
> [5,]  1.5000000 -1.1289063 -2.070313 -0.4003906
> 2.1386719
>>
>
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Thomas Lumley			Assoc. Professor, Biostatistics
tlumley at u.washington.edu	University of Washington, Seattle