For Rotation - KFUPM Faculty List
... and Homogeneous Coordinates •Thus, a general homogeneous coordinate representation can also be written as (h.x, h.y, h). – h can be selected to be any nonzero value. – Thus, there is an infinite number of equivalent homogeneous representations for each coordinate point (x, y). – A convenient choice ...
... and Homogeneous Coordinates •Thus, a general homogeneous coordinate representation can also be written as (h.x, h.y, h). – h can be selected to be any nonzero value. – Thus, there is an infinite number of equivalent homogeneous representations for each coordinate point (x, y). – A convenient choice ...
Non-commutative arithmetic circuits with division
... polynomial size. The complexity of computing polynomials (not allowing division) in noncommuting variables has also been considered, for example, in [35, 25]. This was motivated partly by an apparent lack of progress in proving lower bounds in the commutative setting, partly by an interest in comput ...
... polynomial size. The complexity of computing polynomials (not allowing division) in noncommuting variables has also been considered, for example, in [35, 25]. This was motivated partly by an apparent lack of progress in proving lower bounds in the commutative setting, partly by an interest in comput ...
Secure Distributed Linear Algebra in a Constant Number of
... Jointly Random Secret Invertible Elements and Matrices is a protocol that generates a sharing of a secret, random nonzero field element or an invertible matrix. The protocol securely generates two random elements (matrices), securely multiplies them, and reveals the result. If this is non-zero (inve ...
... Jointly Random Secret Invertible Elements and Matrices is a protocol that generates a sharing of a secret, random nonzero field element or an invertible matrix. The protocol securely generates two random elements (matrices), securely multiplies them, and reveals the result. If this is non-zero (inve ...
Module Fundamentals
... [No other choice is possible for f , and since each fi is a homomorphism, so is f .] We know that every vector space has a basis, but not every module is so fortunate; see Section 4.1, Problem 5. We now examine modules that have this feature. ...
... [No other choice is possible for f , and since each fi is a homomorphism, so is f .] We know that every vector space has a basis, but not every module is so fortunate; see Section 4.1, Problem 5. We now examine modules that have this feature. ...
Pdf - Text of NPTEL IIT Video Lectures
... Geometrically just to give an interpretation of this what actually happens here is that we have now added another coordinate to the point which is we call as W. So the point which was [ex..] 19:34 defined in a plane now has three coordinates x, y, w defined in a space given by x, y, and w. Now if I ...
... Geometrically just to give an interpretation of this what actually happens here is that we have now added another coordinate to the point which is we call as W. So the point which was [ex..] 19:34 defined in a plane now has three coordinates x, y, w defined in a space given by x, y, and w. Now if I ...
nearly associative - American Mathematical Society
... a maximal right ideal which contains no nonzero two-sided ideal of A, and a Jordan algebra is primitive if it contains a maximal inner ideal which contains no nonzero ideal of A. The notion of inner ideal (which will not be defined here) plays the role in Jordan theory of the one-sided ideals in ass ...
... a maximal right ideal which contains no nonzero two-sided ideal of A, and a Jordan algebra is primitive if it contains a maximal inner ideal which contains no nonzero ideal of A. The notion of inner ideal (which will not be defined here) plays the role in Jordan theory of the one-sided ideals in ass ...
Algebraic Numbers and Algebraic Integers
... n, and K̄ be an algebraic closure of K. There are n distinct K-monomorphisms of L into K̄. Proof. This proof is done in two steps. In the first step, the claim is proved in the case when L = K(α), α ∈ L. The second step is a proof by induction on the degree of the extension L/K in the general case, ...
... n, and K̄ be an algebraic closure of K. There are n distinct K-monomorphisms of L into K̄. Proof. This proof is done in two steps. In the first step, the claim is proved in the case when L = K(α), α ∈ L. The second step is a proof by induction on the degree of the extension L/K in the general case, ...
Continuous operators on Hilbert spaces
... [6.3.2] Remark: With σ-finiteness, the argument above is correct whether K is measurable with respect to the product sigma-algebra or only with respect to the completion. ...
... [6.3.2] Remark: With σ-finiteness, the argument above is correct whether K is measurable with respect to the product sigma-algebra or only with respect to the completion. ...
Jordan normal form
In linear algebra, a Jordan normal form (often called Jordan canonical form)of a linear operator on a finite-dimensional vector space is an upper triangular matrix of a particular form called a Jordan matrix, representing the operator with respect to some basis. Such matrix has each non-zero off-diagonal entry equal to 1, immediately above the main diagonal (on the superdiagonal), and with identical diagonal entries to the left and below them. If the vector space is over a field K, then a basis with respect to which the matrix has the required form exists if and only if all eigenvalues of the matrix lie in K, or equivalently if the characteristic polynomial of the operator splits into linear factors over K. This condition is always satisfied if K is the field of complex numbers. The diagonal entries of the normal form are the eigenvalues of the operator, with the number of times each one occurs being given by its algebraic multiplicity.If the operator is originally given by a square matrix M, then its Jordan normal form is also called the Jordan normal form of M. Any square matrix has a Jordan normal form if the field of coefficients is extended to one containing all the eigenvalues of the matrix. In spite of its name, the normal form for a given M is not entirely unique, as it is a block diagonal matrix formed of Jordan blocks, the order of which is not fixed; it is conventional to group blocks for the same eigenvalue together, but no ordering is imposed among the eigenvalues, nor among the blocks for a given eigenvalue, although the latter could for instance be ordered by weakly decreasing size.The Jordan–Chevalley decomposition is particularly simple with respect to a basis for which the operator takes its Jordan normal form. The diagonal form for diagonalizable matrices, for instance normal matrices, is a special case of the Jordan normal form.The Jordan normal form is named after Camille Jordan.