The Fibonacci numbers Consider a population of creatures. Every month, each creature older than one month reproduces, creating one new creature. How does the population grow? � 1 � � � � � pop. at n + 1 1 pop. at n = ≥ one month at n + 1 1 0 ≥ one month at n This was why Fibonacci introduced his numbers.
The Fibonacci numbers Consider a population of creatures. Every month, each creature older than one month reproduces, creating one new creature. How does the population grow? � 1 � � � � � pop. at n + 1 1 pop. at n = ≥ one month at n + 1 1 0 ≥ one month at n This was why Fibonacci introduced his numbers. The appearance of them in nature is sometimes explained by the above mechanism.
Powers of matrices
Powers of matrices To understand a linear discrete dynamical system given by A : V → V
Powers of matrices To understand a linear discrete dynamical system given by A : V → V we should compute A n .
Powers of matrices To understand a linear discrete dynamical system given by A : V → V we should compute A n . If A : R n → R n is given by a diagonal matrix, this is easy:
Powers of matrices To understand a linear discrete dynamical system given by A : V → V we should compute A n . If A : R n → R n is given by a diagonal matrix, this is easy: n a n 0 0 0 0 a 1 1 a n 0 0 0 0 a 2 = 2 a n 0 0 0 0 a 3 3
Diagonal matrices A matrix A is diagonal if and only if:
Diagonal matrices A matrix A is diagonal if and only if: For each e i , there is a scalar λ i so that A · e i = λ i · e i
Diagonal matrices A matrix A is diagonal if and only if: For each e i , there is a scalar λ i so that A · e i = λ i · e i E.g. when n = 3,
Diagonal matrices A matrix A is diagonal if and only if: For each e i , there is a scalar λ i so that A · e i = λ i · e i E.g. when n = 3, this would mean 0 0 λ 1 A = 0 λ 2 0 0 0 λ 3
Diagonal matrices
Diagonal matrices It’s almost as good for A to be diagonal in some basis B = { b 1 , b 2 , . . . , b n }
Diagonal matrices It’s almost as good for A to be diagonal in some basis B = { b 1 , b 2 , . . . , b n } Since in this case, we can change basis to B , compute powers of the diagonal matrix [ A ] B , and then change back.
Diagonal matrices It’s almost as good for A to be diagonal in some basis B = { b 1 , b 2 , . . . , b n } Since in this case, we can change basis to B , compute powers of the diagonal matrix [ A ] B , and then change back. Note that A is diagonal in the basis B exactly when A · b i = λ i b i
Powers in other bases If A : R n → R n is given by a matrix (also called A ),
Powers in other bases If A : R n → R n is given by a matrix (also called A ), If B = { b 1 , b 2 , . . . , b n } is a basis,
Powers in other bases If A : R n → R n is given by a matrix (also called A ), If B = { b 1 , b 2 , . . . , b n } is a basis, set B = [ b 1 , b 2 , . . . , b n ],
Powers in other bases If A : R n → R n is given by a matrix (also called A ), If B = { b 1 , b 2 , . . . , b n } is a basis, set B = [ b 1 , b 2 , . . . , b n ], so that [ v ] B = B − 1 · v [ A ] B = B − 1 AB
Powers in other bases If A : R n → R n is given by a matrix (also called A ), If B = { b 1 , b 2 , . . . , b n } is a basis, set B = [ b 1 , b 2 , . . . , b n ], so that [ v ] B = B − 1 · v [ A ] B = B − 1 AB hence A = B [ A ] B B − 1
Powers in other bases Since A = B [ A ] B B − 1
Powers in other bases Since A = B [ A ] B B − 1 We can compute A 2 = B [ A ] B B − 1 B [ A ] B B − 1 = B [ A ] B [ A ] B B − 1 = B [ A ] 2 B B − 1
Powers in other bases Since A = B [ A ] B B − 1 We can compute A 2 = B [ A ] B B − 1 B [ A ] B B − 1 = B [ A ] B [ A ] B B − 1 = B [ A ] 2 B B − 1 More generally, A n = B [ A ] n B B − 1
Powers in other bases Since A = B [ A ] B B − 1 We can compute A 2 = B [ A ] B B − 1 B [ A ] B B − 1 = B [ A ] B [ A ] B B − 1 = B [ A ] 2 B B − 1 More generally, A n = B [ A ] n B B − 1 So if we can find a basis B in which [ A ] B is diagonal,
Powers in other bases Since A = B [ A ] B B − 1 We can compute A 2 = B [ A ] B B − 1 B [ A ] B B − 1 = B [ A ] B [ A ] B B − 1 = B [ A ] 2 B B − 1 More generally, A n = B [ A ] n B B − 1 So if we can find a basis B in which [ A ] B is diagonal, we can compute [ A ] n B ,
Powers in other bases Since A = B [ A ] B B − 1 We can compute A 2 = B [ A ] B B − 1 B [ A ] B B − 1 = B [ A ] B [ A ] B B − 1 = B [ A ] 2 B B − 1 More generally, A n = B [ A ] n B B − 1 So if we can find a basis B in which [ A ] B is diagonal, we can compute [ A ] n B , hence A n .
Eigenvalues and eigenvectors
Eigenvalues and eigenvectors As we saw, A is diagonal in the basis B exactly when A · b i = λ i b i
Eigenvalues and eigenvectors As we saw, A is diagonal in the basis B exactly when A · b i = λ i b i Any vector b with A · b = λ b is called an eigenvector of A .
Eigenvalues and eigenvectors As we saw, A is diagonal in the basis B exactly when A · b i = λ i b i Any vector b with A · b = λ b is called an eigenvector of A . In this case λ is called an eigenvalue of A .
Eigenvalues and eigenvectors
Eigenvalues and eigenvectors Example Consider the identity matrix I .
Eigenvalues and eigenvectors Example Consider the identity matrix I . Every vector is an eigenvector, since I · v = v
Eigenvalues and eigenvectors Example Consider the identity matrix I . Every vector is an eigenvector, since I · v = v They all have eigenvalue 1.
Eigenvalues and eigenvectors Example � 2 � 0 Consider the matrix A = . 0 3
Eigenvalues and eigenvectors Example � 2 � 0 Consider the matrix A = . Since A · e 1 = 2 e 1 and 0 3 A · e 2 = 3 e 2 ,
Eigenvalues and eigenvectors Example � 2 � 0 Consider the matrix A = . Since A · e 1 = 2 e 1 and 0 3 A · e 2 = 3 e 2 , the vectors e 1 , e 2 are eigenvectors.
Eigenvalues and eigenvectors Example � 2 � 0 Consider the matrix A = . Since A · e 1 = 2 e 1 and 0 3 A · e 2 = 3 e 2 , the vectors e 1 , e 2 are eigenvectors. The vectors a e 1 and b e 2 are also eigenvectors, for any scalars a , b .
Eigenvalues and eigenvectors Example � 2 � 0 Consider the matrix A = . Since A · e 1 = 2 e 1 and 0 3 A · e 2 = 3 e 2 , the vectors e 1 , e 2 are eigenvectors. The vectors a e 1 and b e 2 are also eigenvectors, for any scalars a , b . Are there any other eigenvectors?
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