Linear algebra
1
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
οΏ½ΜοΏ½ οΏ½βοΏ½
π£
[π£1β²π£2β² ]=[πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2] [π£1π£2]
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£
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π₯πΉ π₯πEvent βCausalβ subpopulation Fraction thereof
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π₯πΉ π₯πEvent βCausalβ subpopulation Fraction thereof
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π₯πΉ π₯πEvent βCausalβ subpopulation Fraction thereof
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π₯πΉ π₯π
π₯πΉ (π‘+β π‘ )=π₯πΉ (π‘ )+ππ₯ π (π‘ )βππ₯πΉ (π‘ )+πΏπ₯π (π‘ )βπΌπ₯πΉ (π‘ )
Event βCausalβ subpopulation Fraction thereof
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π₯π (π‘+β π‘ )=π₯π (π‘ )+π π₯π (π‘ )+π π₯πΉ (π‘ )βπΏπ₯ π (π‘ )βπΌ π₯πΉ (π‘ )
Linear algebra
7
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
οΏ½ΜοΏ½ οΏ½βοΏ½
π£
[π£1β²π£2β² ]=[πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2] [π£1π£2]
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£
8
Vector
π£
A vector is an arrow. The position of the head in relation to the tail is expressed in terms of a magnitude and direction.
9
A set of vectors
π£
οΏ½βοΏ½οΏ½βοΏ½
π¦οΏ½βοΏ½
10
A set of vectors
π£
οΏ½βοΏ½οΏ½βοΏ½
π¦οΏ½βοΏ½
11
π£
οΏ½βοΏ½οΏ½βοΏ½
π¦
π£+π€
οΏ½βοΏ½
βHead-to-tailβ addition of and produced a resultant vector not belonging to our original set of vectors
A set of vectors vs. a vector spaceThis scaling (doubling length in this example) of produced 2, which belongs to our original set of vectors
A vector space is a set of vectors that is βclosedβ under scaling and vector addition. Neither scaling nor vector addition produces a result not already included in the βspace.β
12
BasisA vector space is a set of vectors that are βclosedβ under scaling and vector addition.
A set of vectors , , . . .
Linear combination: addition of vectors with scalings
Used a set of vectors to prescribe a vector space!
13
Basis set: Canβt remove any vector without changing space
A vector space is a set of vectors that are βclosedβ under scaling and vector addition.
A set of vectors , , . . .
Linear combination: addition of vectors with scalings
Basis for V vector space V2-dimensionalN
S
EW
14
Basis: Coordinate system
Linear combination: addition of vectors with scalings
A vector space is a set of vectors that are βclosedβ under scaling and vector addition.
A set of vectors , , . . .
15
Basis: Coordinate system
Linear combination: addition of vectors with scalings
A vector space is a set of vectors that are βclosedβ under scaling and vector addition.
A set of vectors , , . . .
π£=π£1π1+π£2π2
π1 π2
16
Basis: Coordinate system
Linear combination: addition of vectors with scalings
A vector space is a set of vectors that are βclosedβ under scaling and vector addition.
A set of vectors , , . . .
π£=π£1π1+π£2π2
π1 π2
οΏ½βοΏ½1οΏ½βοΏ½2 π£=π£1 οΏ½βοΏ½1+π£2 οΏ½βοΏ½2
Linear algebra
17
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
οΏ½ΜοΏ½ οΏ½βοΏ½
π£
[π£1β²π£2β² ]=[πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2] [π£1π£2]
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£
18
Operator
π£
οΏ½ΜοΏ½ οΏ½βοΏ½Given a vector, an operator outputs a vector, possibly scaled and/or rotated
A function associates objects from a domain with objects in a codomain, sometimes in terms of elementary arithmetic operations.
19
Linear operators
πΌπ£
οΏ½ΜοΏ½πΌ οΏ½βοΏ½=πΌ οΏ½ΜοΏ½ οΏ½βοΏ½
Scaling Addition
π£ οΏ½βοΏ½π£+π€
οΏ½ΜοΏ½ οΏ½βοΏ½
π£
οΏ½ΜοΏ½π€οΏ½βοΏ½
οΏ½ΜοΏ½ (πΌοΏ½βοΏ½+π½π€ )=πΌ οΏ½ΜοΏ½ οΏ½βοΏ½+π½ οΏ½ΜοΏ½ οΏ½βοΏ½
20
Representing linear operators
οΏ½ΜοΏ½ (πΌοΏ½βοΏ½+π½π€ )=πΌ οΏ½ΜοΏ½ οΏ½βοΏ½+π½ οΏ½ΜοΏ½ οΏ½βοΏ½
π£=π£1π1+π£2π2
π1 π2
ΒΏπ£1 οΏ½ΜοΏ½π1+π£2 οΏ½ΜοΏ½π2π£ β²= οΏ½ΜοΏ½ οΏ½βοΏ½
π£ β²=π£1β² π1+π£2
β² π2
ΒΏπ£1 [( οΏ½ΜοΏ½π1)1π1+ ( οΏ½ΜοΏ½ π1 )2π2 ]ΒΏ οΏ½ΜοΏ½ (π£1π1+π£2π2 )
+π£2 [( οΏ½ΜοΏ½ π2 )1π1+( οΏ½ΜοΏ½π2 )2π2 ]ΒΏπ£1 [πΈ1 , 1π1+πΈ 2 ,1π2 ]
+π£2 [πΈ 1, 2π1+πΈ2 , 2π2 ]
ΒΏ (π£1πΈ1 , 1+π£2πΈ 1, 2 )π1+(π£1πΈ 2, 1+π£2πΈ 2 ,2 )π2
π£1β² π1+π£2β² π2
π£1β²=πΈ 1 ,1π£1+πΈ 1 ,2π£2π£2β²=πΈ2 ,1π£1+πΈ2 , 2π£2
[π£1β²π£2β² ]=[πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2] [π£1π£2]
πΈ1 , 2
21
Representing linear operators
π£=π£1π1+π£2π2
π1 π2
π£ β²=π£1β² π1+π£2
β² π2
π£1β²=πΈ 1 ,1π£1+πΈ 1 ,2π£2π£2β²=πΈ2 ,1π£1+πΈ2 , 2π£2
[π£1β²π£2β² ]=[πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2] [π£1π£2]
οΏ½ΜοΏ½ (πΌοΏ½βοΏ½+π½π€ )=πΌ οΏ½ΜοΏ½ οΏ½βοΏ½+π½ οΏ½ΜοΏ½ οΏ½βοΏ½ π£ β²= οΏ½ΜοΏ½ οΏ½βοΏ½ Abstract action on vector
Relationship between coefficients
Representation in the context of a particular basis
π£ β²β [π£1β²π£2β² ] π£β[π£1π£2]
οΏ½ΜοΏ½β [πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2]
βThe vector v-prime is represented by the column vector v-prime-sub-1, v-prime-sub-2β
βThe operator A is represented by the matrix Aβ
22
Vector transformation algorithm implies matrix multiplication
π£ β²= οΏ½ΜοΏ½ οΏ½βοΏ½
π£ β² β²=οΏ½ΜοΏ½ π£ β²
π£
οΏ½ΜοΏ½ π£ β²β[πΉ1 ,1 πΉ1 , 2
πΉ2 ,1 πΉ2 , 2][ πΈ 1 ,1π£1+πΈ 1 ,2π£2πΈ 2, 1π£1+πΈ 2 ,2π£2 ]
π£β[π£1π£2]οΏ½ΜοΏ½β [πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2]π£1β²=πΈ 1 ,1π£1+πΈ 1 ,2π£2π£2β²=πΈ2 ,1π£1+πΈ2 , 2π£2
ΒΏ [πΉ1, 1 (πΈ 1 ,1π£1+πΈ1 ,2π£2 )+πΉ1 ,2 (πΈ 2 ,1π£1+πΈ 2 ,2π£2 )πΉ2, 1 (πΈ 1 ,1π£1+πΈ1 ,2π£2 )+πΉ2 , 2 (πΈ 2 ,1π£1+πΈ 2 ,2π£2 )]
ΒΏ [ (πΉ1 ,1πΈ1 ,1+πΉ1 ,2πΈ2 , 1 )π£1+ (πΉ1 , 1πΈ 1, 2+πΉ1, 2πΈ 2 ,2 )π£2(πΉ2 ,1πΈ 1 ,1+πΉ2 ,2πΈ2 , 1 )π£1+ (πΉ2 ,1πΈ 1 ,2+πΉ2 ,2πΈ2 , 2 )π£2]
ΒΏ [πΉ1, 1πΈ 1, 1+πΉ1, 2πΈ 2 ,1 πΉ1 , 1πΈ1 , 2+πΉ1 , 2πΈ 2 ,2
πΉ2, 1πΈ 1, 1+πΉ2 ,2πΈ 2 ,1 πΉ2 , 1πΈ1 , 2+πΉ2, 2πΈ 2 ,2][π£1π£2]οΏ½ΜοΏ½ οΏ½ΜοΏ½π£β[πΉ1 ,1 πΉ1 ,2
πΉ2 ,1 πΉ2 ,2] [πΈ1 ,1 πΈ1 , 2
πΈ2 , 1 πΈ2 , 2] [π£1π£2]
Linear algebra
23
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
οΏ½ΜοΏ½ οΏ½βοΏ½
π£
[π£1β²π£2β² ]=[πΈ 1 ,1 πΈ 1 ,2
πΈ 2 ,1 πΈ 2 ,2] [π£1π£2]
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£
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π₯πΉ π₯πEvent βCausalβ subpopulation Fraction thereof
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π₯πΉ π₯πEvent βCausalβ subpopulation Fraction thereof
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π₯πΉ π₯πEvent βCausalβ subpopulation Fraction thereof
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π₯πΉ π₯πEvent βCausalβ subpopulation Fraction thereof
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π₯πΉ π₯π
π₯πΉ (π‘+β π‘ )=π₯πΉ (π‘ )+ππ₯ π (π‘ )βππ₯πΉ (π‘ )+πΏπ₯π (π‘ )βπΌπ₯πΉ (π‘ )
Event βCausalβ subpopulation Fraction thereof
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π₯π (π‘+β π‘ )=π₯π (π‘ )+π π₯π (π‘ )+π π₯πΉ (π‘ )βπΏπ₯ π (π‘ )βπΌ π₯πΉ (π‘ )
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Free Premiumπ₯πΉ π₯ππ₯πΉ (π‘+β π‘ )=π₯πΉ (π‘ )+ππ₯ π (π‘ )βππ₯πΉ (π‘ )+πΏπ₯π (π‘ )βπΌπ₯πΉ (π‘ )π₯π (π‘+β π‘ )=π₯π (π‘ )+π π₯π (π‘ )+π π₯πΉ (π‘ )βπΏπ₯ π (π‘ )βπΌ π₯πΉ (π‘ )
[π₯πΉ (π‘+β π‘ )π₯π (π‘+β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ ][π₯πΉ (π‘ )π₯π (π‘ ) ]
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½βοΏ½ (π‘ )=π₯πΉ (π‘ ) οΏ½βοΏ½ +π₯π (π‘ ) οΏ½βοΏ½
0 0.25 0.5 0.75 1 1.25 1.5 1.750
0.25
0.5
0.75
1
1.25
1.5
1.75 οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
π₯ π
π
π₯ πΉ
π
Easy-
lookin
g-one-d
imen
siona
l pro
blem
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£οΏ½ΜοΏ½ οΏ½βοΏ½=π πΌ οΏ½βοΏ½
οΏ½ΜοΏ½ οΏ½βοΏ½β π οΏ½ΜοΏ½ οΏ½βοΏ½= 0β( οΏ½ΜοΏ½β ππΌ ) οΏ½βοΏ½= 0β
([πΈ1 ,1 πΈ1 , 2
πΈ2 , 1 πΈ2 , 2]β π [1 00 1])[π£πΉ
π£π ]=[00 ]([π ππ π]β π[1 0
0 1 ])[π£ πΉ
π£ π ]=[00]
STOP
Check that
is consistent in a matrix representation
[πβ π ππ πβ π][π£πΉ
π£ π ]=[00](πβ π )π£ πΉ+ππ£π=0π π£πΉ+(πβπ )π£ π=0
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£(πβ π ) (πβ π )π£πΉ+ (πβ π )ππ£π=0
ππ π£πΉ+π (πβ π )π£ π=0- [ ][ (πβπ ) (πβ π )βππ ]π£πΉ=0
(πβ π ) (πβ π )βππ=0ππβ ππβ ππ+π2βππ=0
π2β (π+π ) π+(ππβππ )=0
πΒ±=(π+π )Β±β (π+π )2β4 (1 ) (ππβππ )
2 (1 )
πΒ±=(π+π )Β±βπ2+2ππ+π2β4 ππ+4ππ
2
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£πΒ±=
(π+π )Β±βπ2+2ππ+π2β4 ππ+4ππ2
πΒ±=(π+π )Β±β (πβπ )2+4ππ
2
πΒ±=(2βπβπΌβπΏ )Β±β (πΏβπβπΌ )2+4 (π+πΏ )π
2There are 2 possibly special scaling factors. Does each l actually correspond to a special ?
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£πΒ±=
(π+π )Β±β (πβπ )2+4ππ2
There are 2 possibly special scaling factors. Does each l actually correspond to a special ?
[πβ πΒ± ππ πβ πΒ±] [π£ πΉ
Β±
π£ πΒ± ]=[00]
(πβ πΒ±) π£πΉΒ± +ππ£π
Β±=0ππ£ π
Β±= (πΒ±βπ )π£πΉΒ±
π£ πΒ±=
πΒ±βππ π£ πΉ
Β±
π£ πΒ±=
πΌ+πβ πΏΒ±β (πΏβπβπΌ )2+4 (π+πΏ )π2 (π+πΏ )
π£πΉΒ±
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ οΏ½βοΏ½=ππ£πΒ±=
(π+π )Β±β (πβπ )2+4ππ2
π£ πΒ±=
πΌ+πβ πΏΒ±β (πΏβπβπΌ )2+4 (π+πΏ )π2 (π+πΏ )
π£πΉΒ±
πΒ±
There are 2 special scaling factors; each l corresponds to a special vector . Unless something is hokey, they point in different directions and can serve as a basis.
οΏ½βοΏ½+ΒΏ ΒΏ π£β
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±π£ π
Β±=πΒ±π£πΉΒ±
οΏ½βοΏ½ (π‘ )=π₯πΉ (π‘ ) οΏ½βοΏ½ +π₯π (π‘ ) οΏ½βοΏ½π οΏ½βοΏ½ +0 οΏ½βοΏ½=ππ
+ ΒΏβπ£+ΒΏ+ππβπ£βΒΏ ΒΏ
οΏ½βοΏ½+ΒΏ ΒΏ π£β
οΏ½βοΏ½=π+ΒΏ ΒΏΒΏ
Inaugural trials
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½βοΏ½=π+ΒΏ ΒΏΒΏ
οΏ½βοΏ½=ΒΏπ+ΒΏ+πβ=1ΒΏ π
+ΒΏπ +ΒΏ+π βπβ=0ΒΏ ΒΏ
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±π£ π
Β±=πΒ±π£πΉΒ±
οΏ½βοΏ½ (π‘ )=π₯πΉ (π‘ ) οΏ½βοΏ½ +π₯π (π‘ ) οΏ½βοΏ½π οΏ½βοΏ½ +0 οΏ½βοΏ½=ππ
+ ΒΏβπ£+ΒΏ+ππβπ£βΒΏ ΒΏ
Inaugural trials
οΏ½βοΏ½=π+ΒΏ ΒΏΒΏ
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
π+ΒΏ+πβ=1ΒΏ π+ΒΏπ +ΒΏ+π βπβ=0ΒΏ ΒΏ
π+ΒΏπ +ΒΏ=β πβπβΒΏ ΒΏ
π+ΒΏ=βπβ
πβ
π+ΒΏΒΏΒΏβπβ
πβ
π+ΒΏ+πβ=1ΒΏ
πβΒΏπβΒΏ πβ=
π+ΒΏ
π+ΒΏβπβ
ΒΏΒΏ π
+ΒΏ=β πβ
π+ΒΏβπβ
ΒΏΒΏ
οΏ½βοΏ½ (π‘ )=ππ+ΒΏβ π£+ΒΏ +π πβπ£βΒΏ ΒΏ
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±π£ π
Β±=πΒ±π£πΉΒ±
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
πβ=π+ΒΏ
π+ΒΏβπβ
ΒΏΒΏ π
+ΒΏ=β πβ
π+ΒΏβπβ
ΒΏΒΏ
οΏ½βοΏ½ (π‘ )=ππ+ΒΏβ π£+ΒΏ +π πβπ£βΒΏ ΒΏ
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±π£ π
Β±=πΒ±π£πΉΒ±
οΏ½ΜοΏ½ οΏ½ΜοΏ½ οΏ½ΜοΏ½ οΏ½βοΏ½ (π‘ )=β ππβ
π+ΒΏβπβ
π+ΒΏπ+ΒΏ π+ ΒΏ οΏ½ΜοΏ½β
π£ +ΒΏ+ ππ+ΒΏ
π+ ΒΏβπβ πβ πβ πβ οΏ½ΜοΏ½ οΏ½βοΏ½βΒΏΒΏ ΒΏ ΒΏ
ΒΏ ΒΏΒΏ
οΏ½ΜοΏ½π οΏ½βοΏ½ (π‘ )=β ππβ
π+ΒΏβπβ
π+ΒΏπ π£+ΒΏ + ππ+ΒΏ
π+ΒΏβπβ
πβππ£βΒΏ
ΒΏΒΏ ΒΏΒΏ
Example: Modeling a freemium cloud data storage business
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±π£ π
Β±=πΒ±π£πΉΒ±
οΏ½ΜοΏ½π οΏ½βοΏ½ (π‘ )=β ππβ
π+ΒΏβπβ
π+ΒΏπ π£+ΒΏ + ππ+ΒΏ
π+ΒΏβπβ
πβππ£βΒΏ
ΒΏΒΏ ΒΏΒΏ
οΏ½βοΏ½ (π‘+πβ π‘ )=β ππβ
π+ΒΏβπβ
π+ΒΏπΒΏ ΒΏΒΏ
π₯πΉ (π‘+πβ π‘ )=ππ+ΒΏπβ
πβπβ π+ΒΏ π
π +ΒΏ βπβ
ΒΏΒΏ
ΒΏ
π₯π (π‘+π β π‘ )=π π+ΒΏπβ
π+ΒΏβπβ
ΒΏΒΏΒΏ
Example: Modeling a freemium cloud data storage business
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±π£ π
Β±=πΒ±π£πΉΒ±
π₯πΉ (π‘+πβ π‘ )=ππ+ΒΏπβ
πβπβ π+ΒΏ π
π +ΒΏ βπβ
ΒΏΒΏ
ΒΏ
π₯π (π‘+π β π‘ )=π π+ΒΏπβ
π+ΒΏβπβ
ΒΏΒΏΒΏ
πΒ±=(2βπβπΌβπΏ )Β±β (πΏβπβπΌ )2+4 (π+πΏ )π
2
πΒ±=πΌ+πβπΏΒ±β (πΏβπβπΌ )2+4 (π+πΏ )π
2 (π+πΏ )
, , , = 0.2, 0.2, 0.1, 0.1
0 0.25 0.5 0.75 1 1.25 1.5 1.750
0.25
0.5
0.75
1
1.25
1.5
1.75
Example: Modeling a freemium cloud data storage business
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
π₯ π
π
π₯ πΉ
π
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±
, , , = 0.2, 0.2, 0.1, 0.1
Example: Modeling a freemium cloud data storage business
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π₯πΉ
π₯π
[π₯πΉ (π‘+πβ π‘ )π₯π (π‘+π β π‘ )]=[1βπβπΌ π+πΏ
π 1βπΏ][1βπβπΌ π+πΏπ 1βπΏ]β― [1βπβπΌ π+πΏ
π 1βπΏ ] [π₯ πΉ (π‘ )π₯ π (π‘ )]
M copies of matrix
οΏ½ΜοΏ½ π£Β±=πΒ± π£Β±π£ π
Β±=πΒ±π£πΉΒ±
π₯πΉ (π‘+πβ π‘ )=ππ+ΒΏπβ
πβπβ π+ΒΏ π
π +ΒΏ βπβ
ΒΏΒΏ
ΒΏ
π₯π (π‘+π β π‘ )=π π+ΒΏπβ
π+ΒΏβπβ
ΒΏΒΏΒΏ
πΒ±=(2βπβπΌβπΏ )Β±β (πΏβπβπΌ )2+4 (π+πΏ )π
2
πΒ±=πΌ+πβπΏΒ±β (πΏβπβπΌ )2+4 (π+πΏ )π
2 (π+πΏ )
Eigenvectors
Eigenvalues
, , , = 0.2, 0.2, 0.1, 0.1