Letter Message
23 September 2010
Letter Message
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Encoding a seven letter message in a matrix?
The message is CALL_ME and it is 7 letters long including the space between call and me. how can i encode it? Can you do a 2x2 matrices or not? Each letter of the alphabet is assigned a number.
_=0
A=1
C=3
E=5
L=12
M=13
Please explain and show your work.
Cool question, although I haven't done this before, I'll give you my approach.
You can write the problem as follows:
y = ax
Where 'y' is a matrix (actually a vector) representing the encoded message. 'a' is a matrix representing the encoding process. And 'x' is the original message (a vector).
If you want the encoder to be a 2x2 matrix, then that forces you to make 'x' into a 2xn vector (and gives you 'y' as a 2xn vector).
So let's say that the encoder is a 2x2 matrix. Then you have to write out the message (in terms of numbers) as:
x =
[3, 1, 12, 12]
[0, 13, 5, -1]
That's a 2x4 matrix by the way, which is the message (CALL_ME) written in terms of numbers. I've included the number -1, which corresponds to a sort of 'ignore' symbol (it's not a letter or a space, its needs to be there to fill out the matrix).
So now you can just make up any matrix 'a' (2x2), as long as it is invertible. And that represents your encoder. So for example:
a=
[1, 17]
[7, 9]
Could be an encoder matrix. And now, all you would have to do to encode the message is multiply 'x' (the message) by 'a' (the encoder). Which will give you the encoded message 'y'.
If you have an encoded message, then you can get back the original message by multiplying 'y' (on the left) by the inverse of 'a'.
I think there would be special matrices that would allow you to convert the encoded message directly into numbers which correspond to letters (ie. the encoded message wouldn't have any numbers greater than 26 in it). I assume these would be 'orthonormal' matrices, but I'm not really sure.
