Linear algebra
description
Transcript of Linear algebra
Linear algebra
1
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
�̂� �⃑�
𝑣
[𝑣1′𝑣2′ ]=[𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2] [𝑣1𝑣2]
�̂� �⃑�=𝜆𝑣
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𝑥𝐹 𝑥𝑃
𝑥𝐹 (𝑡+∆ 𝑡 )=𝑥𝐹 (𝑡 )+𝜌𝑥 𝑃 (𝑡 )−𝜐𝑥𝐹 (𝑡 )+𝛿𝑥𝑃 (𝑡 )−𝛼𝑥𝐹 (𝑡 )
Event “Causal” subpopulation Fraction thereof
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𝑥𝑃 (𝑡+∆ 𝑡 )=𝑥𝑃 (𝑡 )+𝜌 𝑥𝑃 (𝑡 )+𝜐 𝑥𝐹 (𝑡 )−𝛿𝑥 𝑃 (𝑡 )−𝛼 𝑥𝐹 (𝑡 )
Linear algebra
7
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
�̂� �⃑�
𝑣
[𝑣1′𝑣2′ ]=[𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2] [𝑣1𝑣2]
�̂� �⃑�=𝜆𝑣
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Vector
𝑣
A vector is an arrow. The position of the head in relation to the tail is expressed in terms of a magnitude and direction.
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A set of vectors
𝑣
�⃑��⃑�
𝑦�⃑�
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A set of vectors
𝑣
�⃑��⃑�
𝑦�⃑�
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𝑣
�⃑��⃑�
𝑦
𝑣+𝑤
�⃑�
“Head-to-tail” 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 “closed” under scaling and vector addition. Neither scaling nor vector addition produces a result not already included in the “space.”
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BasisA vector space is a set of vectors that are “closed” 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!
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Basis set: Can’t remove any vector without changing space
A vector space is a set of vectors that are “closed” 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
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Basis: Coordinate system
Linear combination: addition of vectors with scalings
A vector space is a set of vectors that are “closed” under scaling and vector addition.
A set of vectors , , . . .
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Basis: Coordinate system
Linear combination: addition of vectors with scalings
A vector space is a set of vectors that are “closed” under scaling and vector addition.
A set of vectors , , . . .
𝑣=𝑣1𝑏1+𝑣2𝑏2
𝑏1 𝑏2
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Basis: Coordinate system
Linear combination: addition of vectors with scalings
A vector space is a set of vectors that are “closed” under scaling and vector addition.
A set of vectors , , . . .
𝑣=𝑣1𝑏1+𝑣2𝑏2
𝑏1 𝑏2
�⃑�1�⃑�2 𝑣=𝑣1 �⃑�1+𝑣2 �⃑�2
Linear algebra
17
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
�̂� �⃑�
𝑣
[𝑣1′𝑣2′ ]=[𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2] [𝑣1𝑣2]
�̂� �⃑�=𝜆𝑣
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Operator
𝑣
�̂� �⃑�Given 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.
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Linear operators
𝛼𝑣
�̂�𝛼 �⃑�=𝛼 �̂� �⃑�
Scaling Addition
𝑣 �⃑�𝑣+𝑤
�̂� �⃑�
𝑣
�̂�𝑤�⃑�
�̂� (𝛼�⃑�+𝛽𝑤 )=𝛼 �̂� �⃑�+𝛽 �̂� �⃑�
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Representing linear operators
�̂� (𝛼�⃑�+𝛽𝑤 )=𝛼 �̂� �⃑�+𝛽 �̂� �⃑�
𝑣=𝑣1𝑏1+𝑣2𝑏2
𝑏1 𝑏2
¿𝑣1 �̂�𝑏1+𝑣2 �̂�𝑏2𝑣 ′= �̂� �⃑�
𝑣 ′=𝑣1′ 𝑏1+𝑣2
′ 𝑏2
¿𝑣1 [( �̂�𝑏1)1𝑏1+ ( �̂� 𝑏1 )2𝑏2 ]¿ �̂� (𝑣1𝑏1+𝑣2𝑏2 )
+𝑣2 [( �̂� 𝑏2 )1𝑏1+( �̂�𝑏2 )2𝑏2 ]¿𝑣1 [𝔸1 , 1𝑏1+𝔸 2 ,1𝑏2 ]
+𝑣2 [𝔸 1, 2𝑏1+𝔸2 , 2𝑏2 ]
¿ (𝑣1𝔸1 , 1+𝑣2𝔸 1, 2 )𝑏1+(𝑣1𝔸 2, 1+𝑣2𝔸 2 ,2 )𝑏2
𝑣1′ 𝑏1+𝑣2′ 𝑏2
𝑣1′=𝔸 1 ,1𝑣1+𝔸 1 ,2𝑣2𝑣2′=𝔸2 ,1𝑣1+𝔸2 , 2𝑣2
[𝑣1′𝑣2′ ]=[𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2] [𝑣1𝑣2]
𝔸1 , 2
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Representing linear operators
𝑣=𝑣1𝑏1+𝑣2𝑏2
𝑏1 𝑏2
𝑣 ′=𝑣1′ 𝑏1+𝑣2
′ 𝑏2
𝑣1′=𝔸 1 ,1𝑣1+𝔸 1 ,2𝑣2𝑣2′=𝔸2 ,1𝑣1+𝔸2 , 2𝑣2
[𝑣1′𝑣2′ ]=[𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2] [𝑣1𝑣2]
�̂� (𝛼�⃑�+𝛽𝑤 )=𝛼 �̂� �⃑�+𝛽 �̂� �⃑� 𝑣 ′= �̂� �⃑� Abstract action on vector
Relationship between coefficients
Representation in the context of a particular basis
𝑣 ′→ [𝑣1′𝑣2′ ] 𝑣→[𝑣1𝑣2]
�̂�→ [𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2]
“The vector v-prime is represented by the column vector v-prime-sub-1, v-prime-sub-2”
“The operator A is represented by the matrix A”
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Vector transformation algorithm implies matrix multiplication
𝑣 ′= �̂� �⃑�
𝑣 ′ ′=�̂� 𝑣 ′
𝑣
�̂� 𝑣 ′→[𝔹1 ,1 𝔹1 , 2
𝔹2 ,1 𝔹2 , 2][ 𝔸 1 ,1𝑣1+𝔸 1 ,2𝑣2𝔸 2, 1𝑣1+𝔸 2 ,2𝑣2 ]
𝑣→[𝑣1𝑣2]�̂�→ [𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2]𝑣1′=𝔸 1 ,1𝑣1+𝔸 1 ,2𝑣2𝑣2′=𝔸2 ,1𝑣1+𝔸2 , 2𝑣2
¿ [𝔹1, 1 (𝔸 1 ,1𝑣1+𝔸1 ,2𝑣2 )+𝔹1 ,2 (𝔸 2 ,1𝑣1+𝔸 2 ,2𝑣2 )𝔹2, 1 (𝔸 1 ,1𝑣1+𝔸1 ,2𝑣2 )+𝔹2 , 2 (𝔸 2 ,1𝑣1+𝔸 2 ,2𝑣2 )]
¿ [ (𝔹1 ,1𝔸1 ,1+𝔹1 ,2𝔸2 , 1 )𝑣1+ (𝔹1 , 1𝔸 1, 2+𝔹1, 2𝔸 2 ,2 )𝑣2(𝔹2 ,1𝔸 1 ,1+𝔹2 ,2𝔸2 , 1 )𝑣1+ (𝔹2 ,1𝔸 1 ,2+𝔹2 ,2𝔸2 , 2 )𝑣2]
¿ [𝔹1, 1𝔸 1, 1+𝔹1, 2𝔸 2 ,1 𝔹1 , 1𝔸1 , 2+𝔹1 , 2𝔸 2 ,2
𝔹2, 1𝔸 1, 1+𝔹2 ,2𝔸 2 ,1 𝔹2 , 1𝔸1 , 2+𝔹2, 2𝔸 2 ,2][𝑣1𝑣2]�̂� �̂�𝑣→[𝔹1 ,1 𝔹1 ,2
𝔹2 ,1 𝔹2 ,2] [𝔸1 ,1 𝔸1 , 2
𝔸2 , 1 𝔸2 , 2] [𝑣1𝑣2]
Linear algebra
23
Linear operators and representations
Motivating example: Web start-up
Vector space and basis
Eigenvector-eigenvalue analysis
+
�̂� �⃑�
𝑣
[𝑣1′𝑣2′ ]=[𝔸 1 ,1 𝔸 1 ,2
𝔸 2 ,1 𝔸 2 ,2] [𝑣1𝑣2]
�̂� �⃑�=𝜆𝑣
Example: Modeling a freemium cloud data storage business
24
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𝑥𝐹 𝑥𝑃Event “Causal” subpopulation Fraction thereof
Recruit Premium users +1 0
+ +
+
Example: Modeling a freemium cloud data storage business
25
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𝑥𝐹 𝑥𝑃Event “Causal” subpopulation Fraction thereof
Recruit Premium users +1 0
Upgrade Free users -1 +1
Example: Modeling a freemium cloud data storage business
26
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𝑥𝐹 𝑥𝑃Event “Causal” subpopulation Fraction thereof
Recruit Premium users +1 0
Upgrade Free users -1 +1
Downgrade Premium users +1 -1
Example: Modeling a freemium cloud data storage business
27
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𝑥𝐹 𝑥𝑃Event “Causal” subpopulation Fraction thereof
Recruit Premium users +1 0
Upgrade Free users -1 +1
Downgrade Premium users +1 -1
Attrition Free users -1 0
Example: Modeling a freemium cloud data storage business
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𝑥𝐹 𝑥𝑃
𝑥𝐹 (𝑡+∆ 𝑡 )=𝑥𝐹 (𝑡 )+𝜌𝑥 𝑃 (𝑡 )−𝜐𝑥𝐹 (𝑡 )+𝛿𝑥𝑃 (𝑡 )−𝛼𝑥𝐹 (𝑡 )
Event “Causal” subpopulation Fraction thereof
Recruit Premium users +1 0
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Attrition Free users -1 0
𝑥𝑃 (𝑡+∆ 𝑡 )=𝑥𝑃 (𝑡 )+𝜌 𝑥𝑃 (𝑡 )+𝜐 𝑥𝐹 (𝑡 )−𝛿𝑥 𝑃 (𝑡 )−𝛼 𝑥𝐹 (𝑡 )
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Free Premium𝑥𝐹 𝑥𝑃𝑥𝐹 (𝑡+∆ 𝑡 )=𝑥𝐹 (𝑡 )+𝜌𝑥 𝑃 (𝑡 )−𝜐𝑥𝐹 (𝑡 )+𝛿𝑥𝑃 (𝑡 )−𝛼𝑥𝐹 (𝑡 )𝑥𝑃 (𝑡+∆ 𝑡 )=𝑥𝑃 (𝑡 )+𝜌 𝑥𝑃 (𝑡 )+𝜐 𝑥𝐹 (𝑡 )−𝛿𝑥 𝑃 (𝑡 )−𝛼 𝑥𝐹 (𝑡 )
[𝑥𝐹 (𝑡+∆ 𝑡 )𝑥𝑃 (𝑡+∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ][𝑥𝐹 (𝑡 )𝑥𝑃 (𝑡 ) ]
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�⃑� (𝑡 )=𝑥𝐹 (𝑡 ) �⃑� +𝑥𝑃 (𝑡 ) �⃑�
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 �̂� �⃑�=𝜆𝑣
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
𝑥 𝑃
𝑁
𝑥 𝐹
𝑁
Easy-
lookin
g-one-d
imen
siona
l pro
blem
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� �⃑�=𝜆𝑣�̂� �⃑�=𝜆 𝐼 �⃑�
�̂� �⃑�− 𝜆 �̂� �⃑�= 0⃑( �̂�− 𝜆𝐼 ) �⃑�= 0⃑
([𝔸1 ,1 𝔸1 , 2
𝔸2 , 1 𝔸2 , 2]− 𝜆 [1 00 1])[𝑣𝐹
𝑣𝑃 ]=[00 ]([𝑎 𝑏𝑐 𝑑]− 𝜆[1 0
0 1 ])[𝑣 𝐹
𝑣 𝑃 ]=[00]
STOP
Check that
is consistent in a matrix representation
[𝑎− 𝜆 𝑏𝑐 𝑑− 𝜆][𝑣𝐹
𝑣 𝑃 ]=[00](𝑎− 𝜆 )𝑣 𝐹+𝑏𝑣𝑃=0𝑐 𝑣𝐹+(𝑑−𝜆 )𝑣 𝑃=0
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� �⃑�=𝜆𝑣(𝑑− 𝜆 ) (𝑎− 𝜆 )𝑣𝐹+ (𝑑− 𝜆 )𝑏𝑣𝑃=0
𝑏𝑐 𝑣𝐹+𝑏 (𝑑− 𝜆 )𝑣 𝑃=0- [ ][ (𝑎−𝜆 ) (𝑑− 𝜆 )−𝑏𝑐 ]𝑣𝐹=0
(𝑎− 𝜆 ) (𝑑− 𝜆 )−𝑏𝑐=0𝑎𝑑− 𝜆𝑎− 𝜆𝑑+𝜆2−𝑏𝑐=0
𝜆2− (𝑎+𝑑 ) 𝜆+(𝑎𝑑−𝑏𝑐 )=0
𝜆±=(𝑎+𝑑 )±√ (𝑎+𝑑 )2−4 (1 ) (𝑎𝑑−𝑏𝑐 )
2 (1 )
𝜆±=(𝑎+𝑑 )±√𝑎2+2𝑎𝑑+𝑑2−4 𝑎𝑑+4𝑏𝑐
2
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� �⃑�=𝜆𝑣𝜆±=
(𝑎+𝑑 )±√𝑎2+2𝑎𝑑+𝑑2−4 𝑎𝑑+4𝑏𝑐2
𝜆±=(𝑎+𝑑 )±√ (𝑎−𝑑 )2+4𝑏𝑐
2
𝜆±=(2−𝜐−𝛼−𝛿 )±√ (𝛿−𝜐−𝛼 )2+4 (𝜌+𝛿 )𝜐
2There are 2 possibly special scaling factors. Does each l actually correspond to a special ?
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� �⃑�=𝜆𝑣𝜆±=
(𝑎+𝑑 )±√ (𝑎−𝑑 )2+4𝑏𝑐2
There are 2 possibly special scaling factors. Does each l actually correspond to a special ?
[𝑎− 𝜆± 𝑏𝑐 𝑑− 𝜆±] [𝑣 𝐹
±
𝑣 𝑃± ]=[00]
(𝑎− 𝜆±) 𝑣𝐹± +𝑏𝑣𝑃
±=0𝑏𝑣 𝑃
±= (𝜆±−𝑎 )𝑣𝐹±
𝑣 𝑃±=
𝜆±−𝑎𝑏 𝑣 𝐹
±
𝑣 𝑃±=
𝛼+𝜐− 𝛿±√ (𝛿−𝜐−𝛼 )2+4 (𝜌+𝛿 )𝜐2 (𝜌+𝛿 )
𝑣𝐹±
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� �⃑�=𝜆𝑣𝜆±=
(𝑎+𝑑 )±√ (𝑎−𝑑 )2+4𝑏𝑐2
𝑣 𝑃±=
𝛼+𝜐− 𝛿±√ (𝛿−𝜐−𝛼 )2+4 (𝜌+𝛿 )𝜐2 (𝜌+𝛿 )
𝑣𝐹±
𝑄±
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.
�⃑�+¿ ¿ 𝑣−
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� 𝑣±=𝜆± 𝑣±𝑣 𝑃
±=𝑄±𝑣𝐹±
�⃑� (𝑡 )=𝑥𝐹 (𝑡 ) �⃑� +𝑥𝑃 (𝑡 ) �⃑�𝑁 �⃑� +0 �⃑�=𝑁𝑐
+ ¿⃑𝑣+¿+𝑁𝑐−𝑣−¿ ¿
�⃑�+¿ ¿ 𝑣−
�⃑�=𝑐+¿ ¿¿
Inaugural trials
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�⃑�=𝑐+¿ ¿¿
�⃑�=¿𝑐+¿+𝑐−=1¿ 𝑐
+¿𝑄 +¿+𝑐 −𝑄−=0¿ ¿
�̂� 𝑣±=𝜆± 𝑣±𝑣 𝑃
±=𝑄±𝑣𝐹±
�⃑� (𝑡 )=𝑥𝐹 (𝑡 ) �⃑� +𝑥𝑃 (𝑡 ) �⃑�𝑁 �⃑� +0 �⃑�=𝑁𝑐
+ ¿⃑𝑣+¿+𝑁𝑐−𝑣−¿ ¿
Inaugural trials
�⃑�=𝑐+¿ ¿¿
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
𝑐+¿+𝑐−=1¿ 𝑐+¿𝑄 +¿+𝑐 −𝑄−=0¿ ¿
𝑐+¿𝑄 +¿=− 𝑐−𝑄−¿ ¿
𝑐+¿=−𝑐−
𝑄−
𝑄+¿¿¿−𝑐−
𝑄−
𝑄+¿+𝑐−=1¿
𝑐−¿𝑐−¿ 𝑐−=
𝑄+¿
𝑄+¿−𝑄−
¿¿ 𝑐
+¿=− 𝑄−
𝑄+¿−𝑄−
¿¿
�⃑� (𝑡 )=𝑁𝑐+¿⃑ 𝑣+¿ +𝑁 𝑐−𝑣−¿ ¿
�̂� 𝑣±=𝜆± 𝑣±𝑣 𝑃
±=𝑄±𝑣𝐹±
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
𝑐−=𝑄+¿
𝑄+¿−𝑄−
¿¿ 𝑐
+¿=− 𝑄−
𝑄+¿−𝑄−
¿¿
�⃑� (𝑡 )=𝑁𝑐+¿⃑ 𝑣+¿ +𝑁 𝑐−𝑣−¿ ¿
�̂� 𝑣±=𝜆± 𝑣±𝑣 𝑃
±=𝑄±𝑣𝐹±
�̂� �̂� �̂� �⃑� (𝑡 )=− 𝑁𝑄−
𝑄+¿−𝑄−
𝜆+¿𝜆+¿ 𝜆+ ¿ �̂�⃑
𝑣 +¿+ 𝑁𝑄+¿
𝑄+ ¿−𝑄− 𝜆− 𝜆− 𝜆− �̂� �⃑�−¿¿ ¿ ¿
¿ ¿¿
�̂�𝑀 �⃑� (𝑡 )=− 𝑁𝑄−
𝑄+¿−𝑄−
𝜆+¿𝑀 𝑣+¿ + 𝑁𝑄+¿
𝑄+¿−𝑄−
𝜆−𝑀𝑣−¿
¿¿ ¿¿
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� 𝑣±=𝜆± 𝑣±𝑣 𝑃
±=𝑄±𝑣𝐹±
�̂�𝑀 �⃑� (𝑡 )=− 𝑁𝑄−
𝑄+¿−𝑄−
𝜆+¿𝑀 𝑣+¿ + 𝑁𝑄+¿
𝑄+¿−𝑄−
𝜆−𝑀𝑣−¿
¿¿ ¿¿
�⃑� (𝑡+𝑀∆ 𝑡 )=− 𝑁𝑄−
𝑄+¿−𝑄−
𝜆+¿𝑀¿ ¿¿
𝑥𝐹 (𝑡+𝑀∆ 𝑡 )=𝑁𝑄+¿𝜆−
𝑀−𝑄− 𝜆+¿ 𝑀
𝑄 +¿ −𝑄−
¿¿
¿
𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )=𝑁 𝑄+¿𝑄−
𝑄+¿−𝑄−
¿¿¿
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� 𝑣±=𝜆± 𝑣±𝑣 𝑃
±=𝑄±𝑣𝐹±
𝑥𝐹 (𝑡+𝑀∆ 𝑡 )=𝑁𝑄+¿𝜆−
𝑀−𝑄− 𝜆+¿ 𝑀
𝑄 +¿ −𝑄−
¿¿
¿
𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )=𝑁 𝑄+¿𝑄−
𝑄+¿−𝑄−
¿¿¿
𝜆±=(2−𝜐−𝛼−𝛿 )±√ (𝛿−𝜐−𝛼 )2+4 (𝜌+𝛿 )𝜐
2
𝑄±=𝛼+𝜐−𝛿±√ (𝛿−𝜐−𝛼 )2+4 (𝜌+𝛿 )𝜐
2 (𝜌+𝛿 )
, , , = 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
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
𝑥 𝑃
𝑁
𝑥 𝐹
𝑁
�̂� 𝑣±=𝜆± 𝑣±
, , , = 0.2, 0.2, 0.1, 0.1
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𝑥𝐹
𝑥𝑃
[𝑥𝐹 (𝑡+𝑀∆ 𝑡 )𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )]=[1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿][1−𝜐−𝛼 𝜌+𝛿𝜐 1−𝛿]⋯ [1−𝜐−𝛼 𝜌+𝛿
𝜐 1−𝛿 ] [𝑥 𝐹 (𝑡 )𝑥 𝑃 (𝑡 )]
M copies of matrix
�̂� 𝑣±=𝜆± 𝑣±𝑣 𝑃
±=𝑄±𝑣𝐹±
𝑥𝐹 (𝑡+𝑀∆ 𝑡 )=𝑁𝑄+¿𝜆−
𝑀−𝑄− 𝜆+¿ 𝑀
𝑄 +¿ −𝑄−
¿¿
¿
𝑥𝑃 (𝑡+𝑀 ∆ 𝑡 )=𝑁 𝑄+¿𝑄−
𝑄+¿−𝑄−
¿¿¿
𝜆±=(2−𝜐−𝛼−𝛿 )±√ (𝛿−𝜐−𝛼 )2+4 (𝜌+𝛿 )𝜐
2
𝑄±=𝛼+𝜐−𝛿±√ (𝛿−𝜐−𝛼 )2+4 (𝜌+𝛿 )𝜐
2 (𝜌+𝛿 )
Eigenvectors
Eigenvalues
, , , = 0.2, 0.2, 0.1, 0.1