Download Matrix Operations (10/6/04)

Matrix Operations (10/6/04)
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Matrices can be added, scalar multiplied,
and multiplied. The former two are
simple; the last more complicated.
A scalar t times a matrix A is just t times
every entry of A.
A + B means add all pairs of
corresponding entries.
A and B can only be added if they are
the same size (both m by n, say).
Matrix Multiplication
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By “multiplying” two matrices A and B ,
we do not mean multiplying the pairs of
corresponding entries (like addition).
Rather, we mean the operation needed
so that multiplication of matrices
corresponds correctly to composition of
the linear transformations they
represent.
Matrix Multiplication Defined
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By (A B) applied to x then, we mean
(A B) x = A (B x)
= A (x1 b1 + … + xn bn )
= x1 A b1 + … + xn A bn
= (A b1 … A bn ) x
That is, by A B we mean the matrix whose
columns are obtained by A operating on the
columns of B .
Hence if B has size m by n , then A must have
m columns, but can have any number of rows.
More on Matrix Multiplication
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Thus if A is a p by m matrix and B is an
m by n matrix, then the product A B is
defined and is a p by n matrix.
Note that A B may be defined but B A
not defined, depending on their sizes.
In particular, it is not true in general
that A B = B A , even if they are both
defined.
Powers and Transposes
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If A is a square matrix, then by the k th
power of A (denoted A k ) we mean A A … A
k times. By A 0 we simply mean the identity
matrix In .
If A is any matrix (m by n, say), the
transpose of A, denoted AT, is the
n by m matrix whose (j, i) entry is the (i, j)
entry of A.
Assignment for Friday
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Read the introduction to Chapter 2 and
Section 2.1.
Do the Practice and Exercises 1, 3, 5, 7,
10, 11, 15, 25, 27, and 28.