Linear Algebra Rekha Santhanam Johns Hopkins Univ. April 3, 2009 Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 1 / 7
Dynamical Systems Denote owl and wood rat populations at time k by O k and R k respectively. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 2 / 7
Dynamical Systems Denote owl and wood rat populations at time k by O k and R k respectively. The owl preys on the wood rat, so if there are no wood rats the population of owls will go down by 50%. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 2 / 7
Dynamical Systems Denote owl and wood rat populations at time k by O k and R k respectively. The owl preys on the wood rat, so if there are no wood rats the population of owls will go down by 50%. If there are no owls to prey on the rats, then the rat population will increase by 10%. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 2 / 7
Dynamical Systems Denote owl and wood rat populations at time k by O k and R k respectively. The owl preys on the wood rat, so if there are no wood rats the population of owls will go down by 50%. If there are no owls to prey on the rats, then the rat population will increase by 10%. In particular, the rat and owl populations dependence is as follows. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 2 / 7
Dynamical Systems Denote owl and wood rat populations at time k by O k and R k respectively. The owl preys on the wood rat, so if there are no wood rats the population of owls will go down by 50%. If there are no owls to prey on the rats, then the rat population will increase by 10%. In particular, the rat and owl populations dependence is as follows. O k+1 = 0.5O k + 0.4R k R k+1 = po k + 1.1R k Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 2 / 7
Dynamical Systems Denote owl and wood rat populations at time k by O k and R k respectively. The owl preys on the wood rat, so if there are no wood rats the population of owls will go down by 50%. If there are no owls to prey on the rats, then the rat population will increase by 10%. In particular, the rat and owl populations dependence is as follows. O k+1 = 0.5O k + 0.4R k R k+1 = po k + 1.1R k The term p calculates the rats preyed by the owls. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 2 / 7
Dynamical Systems Denote owl and wood rat populations at time k by O k and R k respectively. The owl preys on the wood rat, so if there are no wood rats the population of owls will go down by 50%. If there are no owls to prey on the rats, then the rat population will increase by 10%. In particular, the rat and owl populations dependence is as follows. O k+1 = 0.5O k + 0.4R k R k+1 = po k + 1.1R k The term p calculates the rats preyed by the owls. If we start with a certain initial population of owls and rats, how many will be there in, say, 50 years.? Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 2 / 7
Dynamical Systems Examples More generally, we are trying to solve systems of the form x k+1 = A x k, Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 3 / 7
Dynamical Systems Examples More generally, we are trying to solve systems of the form x k+1 = A x k, where A is a n n matrix and x R n. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 3 / 7
Dynamical Systems Examples More generally, we are trying to solve systems of the form x k+1 = A x k, where A is a n n matrix and x R n. Let us consider the examples when [ ] 2 0 A =. 0 0.5 Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 3 / 7
Dynamical Systems Examples More generally, we are trying to solve systems of the form x k+1 = A x k, where A is a n n matrix and x R n. Let us consider the examples when [ ] 2 0 A =. 0 0.5 [ ] 0.5 0 A =. 0 0.2 Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 3 / 7
Dynamical Systems Examples More generally, we are trying to solve systems of the form x k+1 = A x k, where A is a n n matrix and x R n. Let us consider the examples when [ ] 2 0 A =. 0 0.5 [ ] 0.5 0 A =. 0 0.2 [ ] 2 0 A =. 0 1.5 Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 3 / 7
Let A be a n n matrix. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 4 / 7
Let A be a n n matrix. Then a real number λ is said to be an eigenvalue of A if there exists a non-zero vector x R n such that A x = λ x. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 4 / 7
Let A be a n n matrix. Then a real number λ is said to be an eigenvalue of A if there exists a non-zero vector x R n such that A x = λ x. The non-zero vector x corresponding to λ is called an eigenvector. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 4 / 7
Let A be a n n matrix. Then a real number λ is said to be an eigenvalue of A if there exists a non-zero vector x R n such that A x = λ x. The non-zero vector x corresponding to λ is called an eigenvector. Note there are infinitely many eigenvectors corresponding to λ. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 4 / 7
To compute an eigenvalue λ of A, we note that having a nonzero vector x, A x = λ x. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 5 / 7
To compute an eigenvalue λ of A, we note that having a nonzero vector x, A x = λ x. Then this means we need a λ such that the Kernel of the transformation described by A λi n is non-trivial. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 5 / 7
To compute an eigenvalue λ of A, we note that having a nonzero vector x, A x = λ x. Then this means we need a λ such that the Kernel of the transformation described by A λi n is non-trivial. This implies we want to find λ such that det(a λi n ) = 0. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 5 / 7
To compute an eigenvalue λ of A, we note that having a nonzero vector x, A x = λ x. Then this means we need a λ such that the Kernel of the transformation described by A λi n is non-trivial. This implies we want to find λ such that det(a λi n ) = 0. This is known as the characteristic equations and the solutions to this equation are the eigenvalues of A. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 5 / 7
To compute an eigenvalue λ of A, we note that having a nonzero vector x, A x = λ x. Then this means we need a λ such that the Kernel of the transformation described by A λi n is non-trivial. This implies we want to find λ such that det(a λi n ) = 0. This is known as the characteristic equations and the solutions to this equation are the eigenvalues of A. The algebraic multiplicity of λ is its multplicity as a root of the characteristic equation. Note a matrix A is invertible if and only if 0 is not an eigenvalue of A. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 5 / 7
Example Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 6 / 7
Eigenspace Let A be an n n matrix and λ be an eigenvalue of A. Then the eigenspace corresponding to λ is defined to be Ker (A λi n ). Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 7 / 7
Eigenspace Let A be an n n matrix and λ be an eigenvalue of A. Then the eigenspace corresponding to λ is defined to be Ker (A λi n ). Note any two similar matrices have the same eigenvalues. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 7 / 7
Eigenspace Let A be an n n matrix and λ be an eigenvalue of A. Then the eigenspace corresponding to λ is defined to be Ker (A λi n ). Note any two similar matrices have the same eigenvalues. A matrix A is said to be diagonalizable if it is similar to a diagonal matrix. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 7 / 7
Eigenspace Let A be an n n matrix and λ be an eigenvalue of A. Then the eigenspace corresponding to λ is defined to be Ker (A λi n ). Note any two similar matrices have the same eigenvalues. A matrix A is said to be diagonalizable if it is similar to a diagonal matrix. A n n matrix A is diagonalizable if it has n linearly independent eigenvectors. Rekha Santhanam (Johns Hopkins Univ.) Linear Algebra April 3, 2009 7 / 7