Download Sec 2.1 - UBC Math

Math 221, Section 2.1
(1) Matrix addition
(2) Matrix multiplication
(3) Special matrices
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Matrix sum
If A and B are m × n matrices, then the sum A + B is the m × n
matrix whose entries are the sums of the corresponding entries in
A and B.
The subtraction A − B is defined similarly.
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Scalar multiplication of matrices
If c is a scalar and A is a matrix, then the scalar multiple cA is the
matrix whose entries are c times the corresponding entries in A.
Example,
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Matrix multiplication: motivation
When a matrix B multiplies a vector x, it transforms x into the
vector Bx. If Bx is multiplied in turn by a matrix A, the resulting
vector is A(Bx).
We view A(Bx) as produced from x by a composition of linear
transformations. Our goal is to represent this composite function
as multiplication by a single matrix, denoted by AB, so that
(AB)x = A(Bx).
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Matrix multiplication: example
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Matrix multiplication: example
Previous example: The matrix that
(i) reflects points along the x-axis
(ii) then rotates π/2 radians (90◦ ) clockwise.
We have computed
I
I
0 1
, and the second one is −1
0
0 −1 the matrix of the whole process is −1 0 (computed in Sec
1.8).
the first matrix is
1
0
0 −1
Now check that
0 1
−1 0
1
0
0 −1
=
0 −1
−1 0
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Matrix multiplication: definition
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The number of columns of A must match the number of rows
in B in order for the multiplication AB to be defined.
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Otherwise, we cannot multiply A and B.
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Notice that since the first column of AB is Ab1 ; this column is a
linear combination of the columns of A using the entries in b1 as
weights. A similar statement is true for other columns of AB.
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In example 3 on p.5, the first column of AB is Ab1 , which is
11
2
3
=4
+1
−1
1
−5
I
Write similar arguments for the 2nd and the 3rd columns.
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Warnings
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In general
AB 6= BA
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(Example on p.5) What if we reverse the 2 steps:
(i) first rotates π/2 radians (90◦ ) clockwise.
(ii) then reflects points along the x-axis
The resulting matrix is
I
Also check example 7 in the textbook.
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Warnings
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If AB = AC , then it is not true in general that B = C . (That
is, cancellation law does not hold for matrix multiplication.)
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Example (Sec 2.1, Ex 10)
3 −6
−1
A=
,B=
−1 2
3
1
4
,C=
−3
2
−5
1
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Warnings
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If AB = 0 (the zero matrix), it is not true in general that
A = 0 or B = 0.
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Example (Sec 2.1, Ex 12)
3 −6
2
A=
,B=
−2 4
1
4
2
.
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Appendix: properties
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Some special matrices

0 0
0 0
zero matrix 0 =
, 0 0
0 0
0 0

1
1 0
identity matrix I =
, 0
0 1
0

a 0
a 0
diagonal matrix
, 0 b
0 b
0 0
I
I
I

0
0 
0

0 0
1 0 
0 1

0
0 
c
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Zero and one
I
I
I
∗
∗

∗ ∗ ∗ 
∗ ∗ ∗
0 0
0 0
∗
0 0
=
∗
0 0

0 0
0

0 0 =
0
0 0
0
0
,
0 0
0 0
Zero matrix behaves like 0 in numbers: 0a = a0 = 0.
1 0
0 1
1
4

1 2 3 
4 5 6
I
∗
∗
0
0
0
2
5
1
0
0
3
1 2
=
6
4 5

0 0
1

1 0 =
4
0 1
3
6
,
2 3
5 6
Identity matrix behaves like 1 in numbers: 1a = a1 = a.
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Diagonal matrices
I
2 0
0 3
I
I
=
2 4 6
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Multiplying a diagonal matrix D from the left of A results in
row operations on A by scaling each row by the corresponding
diagonal entry of D.
1 2 3
4 5 6
I
1 2 3
4 5 6


2 0 0
 0 3 0  = 2 6 12
8 15 24
0 0 4
Multiplying a diagonal matrix D from the right of A results in
column operations on A by scaling each column by the
corresponding diagonal entry of D.
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Matrix Powers
Ak = A · · · A (k times)
I
A=
I A2
I
= AA =
2 5
1 3
=
AA2
=
2 5
1 3
2 5
1 3
=
9 25
5 14
9 25
=
5 14
2 5
4
3
A = AA =
=
1 3
or you may try
9 25
9 25
4
2
2
A =A A =
=
5 14
5 14
I A3
I
2 5
1 3
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