matrices - ch. 1.6
TRANSCRIPT
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Section Two: Matrices
Textbook: Ch. 1.6
GOALS OF THIS CHAPTER
- define matrix multiplication
- figure out when matrix multiplication fails
- matrix multiplication as linear combinations
- see properties of sigma notation
- see properties of matrix multiplication
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MULTIPLICATION OF MATRICES
If A is an m x p matrix and B is a p x n matrix, then we can do thefollowing preliminary steps:
Matrix multiplication is NOTHING like
regular multiplication!
Bp x n
STEP ONE: Check to see if the number of columns of thefirst matrix is equal to the number of rows of the secondmatrix. If not, the multiplication doesnt exist.
Am x p
Since these numbers are equal, wemove onto step two.
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MULTIPLICATION OF MATRICES
If A is an m x p matrix and B is a p x n matrix, then we can do thefollowing preliminary steps:
Matrix multiplication is NOTHING like
regular multiplication!
Bp x n
STEP TWO: The other two letters or numbers will tell youthe size of the new matrix AB.
Am x p
Read off these letters from left toright. The matrix AB is m x n.
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MULTIPLICATION OF MATRICES
If steps one and two make sense, then we calculate the entries of ABby doing a summation. If we label the entries of AB, A and B as abij,aij, bij, respectively, then:
a ikab ij = bkjk = 1
p
This just means that the (i,j) entry of AB uses allthe information from row i of A and all theinformation from column j of B. We will revisit this
again later.
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
These couple calculations are the mostimportant. I will show you how to break biggermultiplications down into these ones.
B =
-1
0
10
C =3 -5 01
-1A =
C3 x 1B1 x 3Calculate BC:
C3 x 1B1 x 3 BC is 1 x 1
BC exists
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
BC =-1
0
10
3 -5 0
= 3*(-1) +
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
BC =-1
0
10
3 -5 0
= 3(-1) + (-5)*0 +
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
BC =-1
0
10
3 -5 0
= 3(-1) + (-5)0+ 0*(10)
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
BC =-1
0
10
3 -5 0
= 3(-1) + (-5)0+ 0(10)
= -3
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
These couple calculations are the mostimportant. I will show you how to break biggermultiplications down into these ones.
B =
-1
010
C =3 -5 01-1
A =
Calculate AB:
B1 x 3A2 x 1
B1 x 3A2 x 1 AB is 2 x 3
AB exists
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
AB =
=
3 -5 01
-1
Recall that the (1,1) entry ofAB comes from all theinformation in row one of Aand column one of B.
= 1*3
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
AB =
=
3 -5 01
-1
1*3 1*(-5)
The (1,2) entry of AB comesfrom all the information inrow one of A and column two
of B.
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
AB =
=
3 -5 01
-1
1*3 1*(-5) 1*0
The (1,3) entry of AB comesfrom all the information inrow one of A and column
three of B.
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
AB =
=
3 -5 01
-1
1*3 1*(-5) 1*0
(-1)*3
The (2,1) entry of AB comesfrom all the information inrow two of A and column one
of B.
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
AB =
=
3 -5 01
-1
1*3 1*(-5) 1*0
(-1)*3 -1*(-5)
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
AB =
=
3 -5 01
-1
1*3 1*(-5) 1*0
(-1)*3 -1*(-5) (-1)*0
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
AB =
=
3 -5 01
-1
3 -5 0
-3 5 0
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Ex. 1 Row and column matrices
MULTIPLICATION OF MATRICES
These couple calculations are the mostimportant. I will show you how to break biggermultiplications down into these ones.
B =
-1
010
C =3 -5 01-1
A =
Calculate AC:
C3 x 1A2 x 1
These numbers are not equal, so we say AC is not
defined or AC does not exist.
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Ex. 2 Some larger matrices
MULTIPLICATION OF MATRICES
Next we will break down a larger multiplicationinto a row/column matrix multiplication.
B =A =
Calculate AB:
B3 x 2A2 x 3
1 2 -13 1 4
-2 5
4 -3
2 1
B3 x 2A2 x 3 AB is 2 x 2
AB exists
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Ex. 2 Some larger matrices
MULTIPLICATION OF MATRICES
AB = 1 2 -1
3 1 4
-2 5
4 -3
2 1
1 2 -1 -24
2=
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Ex. 2 Some larger matrices
MULTIPLICATION OF MATRICES
AB = 1 2 -13 1 4
-2 5
4 -3
2 1
1 2 -1 1 2 -1-24
2
5-3
1=
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Ex. 2 Some larger matrices
MULTIPLICATION OF MATRICES
AB = 1 2 -13 1 4
-2 5
4 -3
2 1
1 2 -1 1 2 -1
3 1 4
-24
2
-2
4
2
5-3
1=
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Ex. 2 Some larger matrices
MULTIPLICATION OF MATRICES
AB = 1 2 -13 1 4
-2 5
4 -3
2 1
1 2 -1 1 2 -1
3 1 4 3 1 4
-24
2
-2
4
2
5
-3
1
5-3
1=
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MULTIPLICATION OF MATRICES
Ex. 2 Some larger matrices
1(-2)+2*4+(-1)2 1*5+2(-3)+(-1)1
3(-2)+1*4+4*2 3*5+1(-3)+4*1AB =
4 -2
6 16=
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THE IDENTITY MATRIX
Recall that the identity matrix is the square matrix
with only ones on the main diagonal.
When we multiply numbers, any number multiplied by the number 1 isjust itself.
I1 0
0 1=
When we multiply a matrix by an identity matrix ofthe right size, we get the matrix we started with(provided the multiplication makes sense)!
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THE IDENTITY MATRIX
Lets say that I is a 2x2 matrix, A is a 2x3 matrix
and B is a 3x2 matrix.
When multiplying with I, all you really have to do is check that themultiplication makes sense:
I1 0
0 1= A =
1 2 -1
3 1 4B =
-2 5
4 -3
2 1
Calculate AI:
I2 x 2A2 x 3
AI does not exist
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THE IDENTITY MATRIX
Lets say that I is a 2x2 matrix, A is a 2x3 matrix
and B is a 3x2 matrix.
What about the other way around?
I1 0
0 1= A =
1 2 -1
3 1 4B =
-2 5
4 -3
2 1
Calculate IA:
A2 x 3I2 x 2
A2 x 3I2 x 2 IA is 2 x 3
IA exists
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THE IDENTITY MATRIX
Lets say that I is a 2x2 matrix, A is a 2x3 matrixand B is a 3x2 matrix.
In fact, IA is just equal to A (you should check this)! So provided themultiplication makes sense, multiplying by I is pretty easy.
I1 0
0 1= A =
1 2 -1
3 1 4B =
-2 5
4 -3
2 1
I A = A
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THE IDENTITY MATRIX
Lets say that I is a 2x2 matrix, A is a 2x3 matrixand B is a 3x2 matrix.
What side can we multiply I on when dealing with the matrix B?
I1 0
0 1= A =
1 2 -1
3 1 4B =
-2 5
4 -3
2 1
IB = BBI = B
Since B is 3x2, we mustmultiply by I on the rightfor the multiplication tomake sense!
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MATRIX MULTIPLICATION AS LINEAR
COMBINATIONS
a11x1 + a12x2+ + a1nxn
a21x1 + a22x2+ + a2nxn
am1x1 + am2x2+ + amnxn
x1
x2
xn
a11 a12 a1n
a21 a22 a2n
am1 am2 amn
=
Let A be an mxn matrix and let x be an n-vector (or an nx1 columnmatrix). Consider the multiplication Ax:
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MATRIX MULTIPLICATION AS LINEAR
COMBINATIONS
a11x1 + a12x2+ + a1nxn
a21x1 + a22x2+ + a2nxn
am1x1 + am2x2+ + amnxn
=
We can rewrite the right-hand side of the equality as
a11
a21
am1
a12
a22
am2
a1n
a2n
amn
+ +x1 x2 xn+
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MATRIX MULTIPLICATION AS LINEAR
COMBINATIONS
=
By re-labeling the n-vectors as c1, c2, , cn (the subscript denotesthe column of A), we express the multiplication Axas a linearcombination of the column vectors of the matrix A.
a11
a21
am1
a12
a22
am2
a1n
a2n
amn
+ +x1 x2 xn+Ax =x1c1 + x2c2++ xncn
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MATRIX MULTIPLICATION AS LINEAR
COMBINATIONS
We have shown that the product of an mxn matrix A and a columnvector x is a linear combination of the columns of A.
Linear combinations play an important role in higher linear algebracourses and are a fundamental tool in digital music.
We could generalize this finding to a matrix Athat is mxn and a matrix B that is nxp. Theproduct AB is a linear combination of the columnsof A as well! (see page 69-70 of the text formore info)
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MATRIX MULTIPLICATION AS LINEAR
COMBINATIONS
=
Ex. 4 - Lets see an example of this:
1 0 -3
-1 2 57 10 4
-1
0
10
-31
51
93
-31
51
93
=
1-1
7
02
10
-35
4
+ + 100-1
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SIGMA NOTATION
Let A be an mxn matrix, and let B be an nxp matrix. We have shownthat the product AB is defined and is of size mxp. To figure out the(i,j) entry of the matrix AB, we could use the formula:
a ikab ij = bkjk = 1
p
We call this sigma notation, but how exactly doesit work?
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SIGMA NOTATION
Lets say we want the (1,1) entry of AB. This means we
fill in i=1 and j=1 in our formula:
a1kab11 = bk1k = 1
p
Next, we let the variable k=1 to start, then we add the term k=2, k=3,all the way up to k=p:
= a11b11 + a12b21+ + a1pbp1
k=1 k=2 k=p. . .
ab11
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SIGMA NOTATION
Ex. 6 Sum of the first five squares
- done on overhead
Thm. 7 Facts about sigma notation
- done on overhead
- no proof required
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PROPERTIES OF MATRIX MULTIPLICATION
Thm. 8 Properties of matrix multiplication
Provided that A, B and C are matrices of the right size formultiplication, and that r is any number, we have the following:
(1) A(BC) = (AB)C
(2) A(B+C) = AB+AC
(3) (A+B)C = AC+BC
(4) A(rB) = r(AB)
(5) (AB)
T
= B
T
A
T
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PROPERTIES OF MATRIX MULTIPLICATION
Thm. 8 Properties of matrix multiplication
Proof Done on overhead