Definition: A matrix transformation T : R n → Rm is said to be onto if
Definition: A matrix transformation T : R n → Rm is said to be onto if

... origin parallel to the vector (−2, 1) Eigenvalues of powers of a matrix Theorem 5.1.4: If k is a positive integer, λ is an eigenvalue of a matrix A and x is a corresponding eigenvector, then λk is an eigenvalue of Ak and x is a corresponding eigenvector. Idea: Ax = λx ⇒ A2 x = A(Ax) = A(λx) = λAx = ...
Matrix Methods
Matrix Methods

Final Exam [pdf]
Final Exam [pdf]

... (b) If a 6 × 10 matrix is row equivalent to an echelon matrix with 4 non-zero rows, then the dimension of the null space of A is 2. ...
Your Title Here - World of Teaching
Your Title Here - World of Teaching

Test 2 Review Math 3377 (30 points)
Test 2 Review Math 3377 (30 points)

Defn: A set V together with two operations, called addition and
Defn: A set V together with two operations, called addition and

Eigenvectors
Eigenvectors

LSA - University of Victoria
LSA - University of Victoria

2
2

Physics 3730/6720 – Maple 1b – 1 Linear algebra, Eigenvalues and Eigenvectors
Physics 3730/6720 – Maple 1b – 1 Linear algebra, Eigenvalues and Eigenvectors

... their elements. To do matrix and vector products and see the result you need evalm. (Maple does it, otherwise, but silently.) Multiplication by scalars works with *, but multiplication of matrices with matrices and matrices with vectors requires the special multiplication operator &*. > eigenvals(A) ...
  (Some) Matrices and Determinants 
  (Some) Matrices and Determinants 

Solution of 2x2
Solution of 2x2

... The values of are the eigenvalues of the system. If the quadratic formula discriminant, √ , is positive, the matrix will have two distinct, real roots. If the discriminant is 0, the system has 1 real root. If the discriminant is negative, the system will have two complex roots. The eigenvalues are t ...
(pdf)
(pdf)

CHAPTER 7
CHAPTER 7

Let v denote a column vector of the nilpotent matrix Pi(A)(A − λ iI)ni
Let v denote a column vector of the nilpotent matrix Pi(A)(A − λ iI)ni

Applying transformations in succession Suppose that A and B are 2
Applying transformations in succession Suppose that A and B are 2

Problems:
Problems:

Perron–Frobenius theorem

In linear algebra, the Perron–Frobenius theorem, proved by Oskar Perron (1907) and Georg Frobenius (1912), asserts that a real square matrix with positive entries has a unique largest real eigenvalue and that the corresponding eigenvector can be chosen to have strictly positive components, and also asserts a similar statement for certain classes of nonnegative matrices. This theorem has important applications to probability theory (ergodicity of Markov chains); to the theory of dynamical systems (subshifts of finite type); to economics (Okishio's theorem, Leontief's input-output model); to demography (Leslie population age distribution model), to Internet search engines and even ranking of football teams.