Unformatted text preview: he same elements in corresponding
locations
locations
A = B ⇔ aij = bij
∀i, j Symbol ∀ means “for each and every”,
⇔means “if and only if” 4
2 3 4 = 2
0 3 2 ≠ 4
0 0
3 x 7 y = 4 8 Operations on Matrices Addition and subtraction Multiplication
• Scalar multiplication
• Multiplication of matrices Transposition Inversion 9 Addition and Subtraction of Matrices
Two matrices can be added ⇔ two matrices are of the same dimension.
In this case we say the two matrices are conformable for addition.
In
conformable
Addition rule: add each pair of corresponding elements B = [b ], i = 1..m, j = 1..n
C = A + B = [ a + b ], i = 1..m, j = 1..n
A = aij , i = 1..m, j = 1..n
ij ij 4 9 2 0 4 + 2 9 + 0 6 9 2 1 + 0 7 = 2 + 0 1 + 7 = 2 8 ij Subtraction rule: subtract each pair of corresponding elements 19 3 6 8 19 − 6 3 − 8 13 − 5 2 0 − 1 3 = 2 − 1 0 − 3 = 1 − 3 C = A −B
10 Scalar Multiplication
Scalar multiplication of a matrix refers to multiplying each element of a
Scalar
each
matrix by a (real) number. Number in this context is referred to as scalar.
matrix 3 − 1 21 − 7 7 = 0 35 0 5 11 Conformability for Multiplication
Conformability condition for multiplication: matrix A of dimension m x n is
Conformability
conformable for multiplication with matrix B of dimension k x l if n=k.
conformable
Translation: you can multiply A by B if the number of columns in A is equal
Translation:
to the number of rows in B.
to
The resulting dimension will be m x l
Translation: matrix AxB will have A’s number of rows and B’s number of
Translation:
columns.
columns. a
c b
d 1 Is conformable for multiplication with Is
3 2
4 since the number of columns in A (2) is equal to the
since
number of rows in B (also 2).
number
12 Special Case of Matrix Multiplication:
an Inner Product of Two Vectors
Recap: vectors are special matrices one of whose dimensions is equal to 1
vectors
(i.e. vector columns or row vectors).
(i.e.
Definition: an inner product of two vectors u and v with the equal number
of elements is defined as u ⋅ v = u1v1 + u2 v2 + ... + un vn
of
where u = (u , u ,..., u )
where
1 2 n v = ( v1 , v2 ,..., vn ) Suppose I went shopping and bought 2 loaves of bread at $5 each and
3 packs of potato chips at $2 each.
My consumption vector is:
Q' = [ 2,3]
My price vectors is: P ' = [ $5,$2]
My shopping bill today will be exactly the inner product of P ' and Q '
My Q'⋅P ' = Q1 P + Q2 P2 = 2 × $5 + 3 × $2 = $16
1
13 Matrix Multiplication Rule
Each element cij of C=AB is defined as an inner product of row
Each
matrix A and column j in matrix B. 1 3
A = 2 8 4 0 , B = 5 9 1× 5 + 3 × 9 32 C = AB = 2 × 5 + 8 × 9 = 82 4 × 5 + 0 × 9 20 i in 1. A and B are conformable for
and
multiplication: A’s number of columns
is the same with B’s number of rows
is 2. The resulting C=AB has A’s number of
2.
rows (3) and B’s number of columns (1),
so C is a column vector.
so c11 = a11b11 + a12b21 C’s element in row 1 and column 1 is
C’s
an inner pro...
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 Fall '11
 Kim
 Linear Algebra, Matrices, Inverses

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