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MathGroup Archive 2004

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Re: identity matrix

  • To: mathgroup at smc.vnet.net
  • Subject: [mg49961] Re: identity matrix
  • From: Bill Rowe <readnewsciv at earthlink.net>
  • Date: Sat, 7 Aug 2004 03:52:25 -0400 (EDT)
  • Sender: owner-wri-mathgroup at wolfram.com

On 8/6/04 at 3:09 AM, markbishop at charter.net (Mark) wrote:

>Is the identity matrix strictly a matrix with 1's along the
>principal diagonal (and zeros elsewhere),

Yes.

>or is a matrix which reduces (through row and column operations) to
>the above form also the identity matrix.

No.

To see why this is true consider the value of the first element in a matrix multiplication

That would be Sum[A[[1,j]] I[[1,j]] {j,1,m}] where m is the number of columns. The obvious way to make this Sum be A[[1,1]] is for I[[1,1]] = 1 and I[[1,j]]= 0 for all j != 1. When you add the constraint that each element of the result must match the corresponding element of A, it is easy to see I must have 1's in the diagonal and 0's everywhere else. Any other choice would mean A would not equal A I which by definition means I would not be the identity matrix.
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