1 Reminders from 1-dimensional calculus
... The notion discussed above has an analogue in multivariate calculus. Consider a function of two variables f : D → R with domain D ⊂ R2 . Around any point (x0 , y0 ) ∈ D, f can be approximated by a degree 1 polynomial f (x, y) ≈ f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) =: L(x, y ...
... The notion discussed above has an analogue in multivariate calculus. Consider a function of two variables f : D → R with domain D ⊂ R2 . Around any point (x0 , y0 ) ∈ D, f can be approximated by a degree 1 polynomial f (x, y) ≈ f (x0 , y0 ) + fx (x0 , y0 )(x − x0 ) + fy (x0 , y0 )(y − y0 ) =: L(x, y ...
CHAPTER ONE Matrices and System Equations
... using only row operation III, then C has an LU factorization. The matrix L is unit lower triangular, and if i > j, then lij is the multiple of t he jth row subtracted from the ith row during the reduction process. ...
... using only row operation III, then C has an LU factorization. The matrix L is unit lower triangular, and if i > j, then lij is the multiple of t he jth row subtracted from the ith row during the reduction process. ...
EIGENVALUES OF PARTIALLY PRESCRIBED
... this paper are considered to be monic. If f is a polynomial, d(f ) denotes its degree. If ψ1 | · · · |ψn are invariant factors of a polynomial matrix A(λ) over F[λ], rank A(λ) = n, then we assume ψi = 1, for any i ≤ 0, and ψi = 0, for any i ≥ n + 1. Definition 2.1. Let A, A ∈ Fn×n , B, B ∈ Fn×l . ...
... this paper are considered to be monic. If f is a polynomial, d(f ) denotes its degree. If ψ1 | · · · |ψn are invariant factors of a polynomial matrix A(λ) over F[λ], rank A(λ) = n, then we assume ψi = 1, for any i ≤ 0, and ψi = 0, for any i ≥ n + 1. Definition 2.1. Let A, A ∈ Fn×n , B, B ∈ Fn×l . ...
EECS 275 Matrix Computation
... For any µ ∈ IR that is not an eigenvalue of A, the eigenvectors of (A − µI )−1 are the same as the eigenvectors of A, and the the corresponding eigenvalues are (λj − µ)−1 where λj are the eigenvalues of A Suppose µ is close to an eigenvalue λJ of A, then (λJ − µ)−1 may be much larger than (λj − µ)−1 ...
... For any µ ∈ IR that is not an eigenvalue of A, the eigenvectors of (A − µI )−1 are the same as the eigenvectors of A, and the the corresponding eigenvalues are (λj − µ)−1 where λj are the eigenvalues of A Suppose µ is close to an eigenvalue λJ of A, then (λJ − µ)−1 may be much larger than (λj − µ)−1 ...
Fast direct solvers for elliptic PDEs
... For HSS matrix algebra to be numerically stable, it is critical that the basis matrices Uτ and Vτ be well-conditioned. The gold-standard is to have Uτ and Vτ be orthonormal (i.e. σj (Uτ ) = σj (Vτ ) = 1 for j = 1, 2, . . . , k), and this is commonly enforced. We have decided to instead use interpola ...
... For HSS matrix algebra to be numerically stable, it is critical that the basis matrices Uτ and Vτ be well-conditioned. The gold-standard is to have Uτ and Vτ be orthonormal (i.e. σj (Uτ ) = σj (Vτ ) = 1 for j = 1, 2, . . . , k), and this is commonly enforced. We have decided to instead use interpola ...
Homework # 7 Solutions
... Equate coefficients c1 = 0 c2 + 2c2 = 0 =⇒ c2 = 0. Since c1 = c2 = 0 is the only solution, {p1 , p3 } is linearly dependent. Therefore, by the Spanning Set Theorem, it is a basis for Span{p1 , p2 , p3 }. Note that we could also have used the Wronskian to show that {p1 , p3 } is linearly independent, ...
... Equate coefficients c1 = 0 c2 + 2c2 = 0 =⇒ c2 = 0. Since c1 = c2 = 0 is the only solution, {p1 , p3 } is linearly dependent. Therefore, by the Spanning Set Theorem, it is a basis for Span{p1 , p2 , p3 }. Note that we could also have used the Wronskian to show that {p1 , p3 } is linearly independent, ...
Linear algebra
Linear algebra is the branch of mathematics concerning vector spaces and linear mappings between such spaces. It includes the study of lines, planes, and subspaces, but is also concerned with properties common to all vector spaces.The set of points with coordinates that satisfy a linear equation forms a hyperplane in an n-dimensional space. The conditions under which a set of n hyperplanes intersect in a single point is an important focus of study in linear algebra. Such an investigation is initially motivated by a system of linear equations containing several unknowns. Such equations are naturally represented using the formalism of matrices and vectors.Linear algebra is central to both pure and applied mathematics. For instance, abstract algebra arises by relaxing the axioms of a vector space, leading to a number of generalizations. Functional analysis studies the infinite-dimensional version of the theory of vector spaces. Combined with calculus, linear algebra facilitates the solution of linear systems of differential equations.Techniques from linear algebra are also used in analytic geometry, engineering, physics, natural sciences, computer science, computer animation, and the social sciences (particularly in economics). Because linear algebra is such a well-developed theory, nonlinear mathematical models are sometimes approximated by linear models.