Notes on computational linguistics.pdf - UCLA Department of ...

Stabler - Lx 185/209 2003
Matrix arithmetic review: An m × n matrix A is an array with m rows and n columns.
Let A(i, j) be the element in row i and column j.
1. We can add the m × n matrices A, B to get the m × n matrix A + B = C in which
C(i,j) = A(i, j) + B(i,j).
2. Matrix addition is associative and commutative:
A + (B + C) = (A + B) + C
A + B = B + A
3. For any n × m matrix M there is an n × m matrix M ′ such that M + M ′ = M ′ + M = M, namelythen × m
matrix M ′ such that every M(i,j) = 0.
4. We can multiply an m × n matrix A and a n × p matrix to get an m × p matrix C in which
C(i,j) = n k=1 A(i, k)B(k, j).
This definition is sometimes called the “row by column” rule. To find the value of C(i,j), youaddthe
products of all the elements in row i of A and column j of B. For example,
1
2
4 8
5 2
9
6
0 (1 · 8) + (4 · 2)
=
0 (2 · 8) + (5 · 2)
(1 · 9) + (4 · 6)
(2 · 9) + (5 · 6)
(1 · 0) + (4 · 0)
(2 · 2) + (5 · 0)
Here we see that to find C(1, 1) we sum the products of all the elements row 1 of A times the elements
in column 1 of B. – The number of elements in the rows of A must match the number of elements in the
columns of B or else AB is not defined.
5. Matrix multiplication is associative, but not commutative:
A(BC) = (AB)C
For example, 3 5
2 2
= 3 5
4 4
It is interesting to notice that Lambek’s (1958) composition operator is also associative but not commutative:
(X • Y)• Z) ⇒ X • (Y • Z)
X/Y • Y ⇒ X
Y • X/Y ⇒ X
The connection between the Lambek calculus and matrix algebra is actually a deep one (Parker, 1995).
6. For any m × m matrix M there is an m × m matrix Im such that TIm = ImT = T ,namelythem × m matrix
Im such that every Im(i, i) = 1andforeveryi= j,Im(i, j) = 0.
7. Exercise:
a. Explain why the claims in 2 are obviously true.
b. Do the calculation to prove that my counterexample to commutativity in 5 is true.
c. Explain why 6 is true.
d. Make sure you can use octave or some other system to do the calculations once you know how to do
them by hand:
1% octave
Octave, version 2.0.12 (i686-pc-linux-gnulibc1).
Copyright (C) 1996, 1997, 1998 John W. Eaton.
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