Introduction 1 1. Invariant Subspaces 3 2. Eigenvectors, Eigenvalues and Eigenspaces 11 3. Cyclic Spaces 14 4. Prime and Primary Vectors 15 5. The Cyclic Decomposition Theorem 20 6. Rational and Jordan Canonical Forms 22 7. Similarity 23 8. The Cayley-Hamilton Polynomial (Or: Up With Determinants?) 24 9. Extending The Ground Field 25 9.1. Some Invariances Under Base Extension 25 9.2. Semisimpli...

This paper presents a novel methodology to deal with the problem of trajectory tracking in WMR. This new approach uses the dynamic model of WMR and allows perfect velocities tracking. Generally, the classic control approaches applied to the dynamic model of the robot solve the problem of trajectory tracking in two stages. In the first one is designed a controller that ensures the perfect tracki...

Starting in the late 1960s computer scientists including Dijkstra and Hoare advocated goal-oriented programming and formal derivation of algorithms. The problem was that for loop-based programs, a priori determination of loop-invariants, a prerequisite for developing loops, was a task too complex for any but the simplest of operations. We believe that no practical progress was made in the field...

Exercise 3. So far we have been working with vectors in R2 and R3, but it is important to remember that other objects, like functions, also behave like vectors in the sense that we can add them, subtract them, multiply them by scalars, etc. Calculate the following quantities for the two polynomials p(x) := 5x2 + 4x + 2 and q(x) := 3x2 + 4x + 3, and evaluate the result at the point x = 2: (a) p(...

where Aij = a(i, j) and the domain of the function is just the cross-product of the two index sets. Such a structure is called a matrix. The values Aij are called the elements or entries of the matrix. A sequence of elements with the same first index is called a row of the matrix; similarly, a sequence of elements with the same second index is called a column. The dimension of the matrix specif...

defined for all complex numbers λ, where I denotes the � × � identity matrix. It is not hard to see that a complex number λ is an eigenvalue of A if and only if χA(λ) = 0. We see by direct computation that χA is an �th-order polynomial. Therefore, A has precisely � eigenvalues, thanks to the fundamental theorem of algebra. We can write them as λ1� � � � � λ�, or sometimes more precisely as λ1(A...

2 Espaços Lineares (Vectoriais) 7 2.1 Subespaços lineares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.2 Vectores geradores . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3 Independência linear . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.4 Bases e dimensão de espaços lineares . . . ...

This course provides an introduction to the interplay between linear algebra and di¤erential equations/dynamical systems in continuous time. We …rst introduce linear di¤erential equations in Euclidian space R and on Grassmannian and ‡ag manifolds induced by a single matrix A, with emphasis on characterizations of the constant matrix A from a dynamics point of view. We then introduce linear skew...

For a matrix A = [aij ] m,n i,j=1 ∈ Fm×n, the transpose of A is the matrix A> = [aji] n,m j,i=1 ∈ Fn×m. A square matrix A ∈ Rn×n is called symmetric if aji = aij for all i, j ∈ {1, . . . , n} and is called skew-symmetric or anti-symmetric if aji = −aij for all i, j ∈ {1, . . . , n}. A basis will be denoted B = {u1,u2, . . . ,un} when the ordering of the basis vectors is not important and B = [u...

Complexity/Cost We stated last time that for a system of n equations in n unknowns (represented by an N × N matrix A), both LU and QR-factorization take time proportional to the cube of n (O(n3)). LU-factorization takes less time by a constant factor of around 2 as n gets large, this constant factor is dwarfed by the third power of the matrix dimension, however. Note that the theoretical “best-...