On the number of occurrences of a symbol in words of regular
... pair of matrices T; S, the expression T ¿S means that Tij ¿Sij for every pair of indices i; j. As usual, we consider any vector v as a column vector and denote by v the corresponding row vector. We recall that a non-negative matrix T is called primitive if there exists m∈N such that T m ¿0. The mai ...
... pair of matrices T; S, the expression T ¿S means that Tij ¿Sij for every pair of indices i; j. As usual, we consider any vector v as a column vector and denote by v the corresponding row vector. We recall that a non-negative matrix T is called primitive if there exists m∈N such that T m ¿0. The mai ...
A recursive algorithm for computing Cramer-Rao
... for 8 which successively increases the likelihood function and converges to the maximum likelihood estimator. In a similar manner, our algorithm generates a sequence of tighter and tighter lower bounds on estimator covariance which converges to the actual CR matrix bound. The algorithms given here c ...
... for 8 which successively increases the likelihood function and converges to the maximum likelihood estimator. In a similar manner, our algorithm generates a sequence of tighter and tighter lower bounds on estimator covariance which converges to the actual CR matrix bound. The algorithms given here c ...
Matrix algebra for beginners, Part II linear transformations
... parallelogram bounded by the vectors of the two forces, as in Figure 1. A similar rule holds for velocities and this becomes vital if you ever have to navigate a plane or a boat. Your actual velocity is the parallelogram sum of the velocity determined by your speed and heading and the velocity of th ...
... parallelogram bounded by the vectors of the two forces, as in Figure 1. A similar rule holds for velocities and this becomes vital if you ever have to navigate a plane or a boat. Your actual velocity is the parallelogram sum of the velocity determined by your speed and heading and the velocity of th ...
314K pdf
... There are stochastic, iterative, or gradient flow algorithms that will generically perform computations with fewer operations than predicted by this bound. However, the complexity of such algorithms is difficult to understand, or they require unbounded numbers of operations in the worst case. In any ...
... There are stochastic, iterative, or gradient flow algorithms that will generically perform computations with fewer operations than predicted by this bound. However, the complexity of such algorithms is difficult to understand, or they require unbounded numbers of operations in the worst case. In any ...
الوحدة العاشرة
... 9. The determinant of an n×n matrix A times scalar multiple k equal to kn times the determinant of the matrix A, that is det(kA) = kn det(A). 10. The determinant of the kth power of a matrix A equal to the kth power of the determinant of the matrix A, that is det(Ak) = (det(A)) k. 11. The determina ...
... 9. The determinant of an n×n matrix A times scalar multiple k equal to kn times the determinant of the matrix A, that is det(kA) = kn det(A). 10. The determinant of the kth power of a matrix A equal to the kth power of the determinant of the matrix A, that is det(Ak) = (det(A)) k. 11. The determina ...
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.