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Cramer–Rao Lower Bound for Constrained Complex Parameters
... useful developments of the CRB theory have been presented in later research. The first being a CRB formulation for unconstrained complex parameters given in [2]. This treatment has valuable applications in studying the base-band performance of modern communication systems where the problem of estima ...
... useful developments of the CRB theory have been presented in later research. The first being a CRB formulation for unconstrained complex parameters given in [2]. This treatment has valuable applications in studying the base-band performance of modern communication systems where the problem of estima ...
Matrices - MathWorks
... In general, determinants are not very useful in practical computation because they have atrocious scaling properties. But 2-by-2 determinants can be useful in understanding simple matrix properties. If the determinant of a matrix is positive, then multiplication by that matrix preserves left- or rig ...
... In general, determinants are not very useful in practical computation because they have atrocious scaling properties. But 2-by-2 determinants can be useful in understanding simple matrix properties. If the determinant of a matrix is positive, then multiplication by that matrix preserves left- or rig ...
Matrix Algebra
... Here we have placed the solution for x1 , x2 and x3 within the vector x. The correctness of these values may be confirmed by substituting them into the equations of (4). Elementary Operations with Matrices It is often useful to display the generic element of a matrix together with the symbol for the ...
... Here we have placed the solution for x1 , x2 and x3 within the vector x. The correctness of these values may be confirmed by substituting them into the equations of (4). Elementary Operations with Matrices It is often useful to display the generic element of a matrix together with the symbol for the ...
An Introduction to Linear Algebra
... familiar with linear algebra may find this article to be a good quick review. Those totally unfamiliar with linear algebra should consider spending some time with a linear algebra text. In particular, those by Gilbert Strang are particularly easy to read and understand. Several of the numerical exam ...
... familiar with linear algebra may find this article to be a good quick review. Those totally unfamiliar with linear algebra should consider spending some time with a linear algebra text. In particular, those by Gilbert Strang are particularly easy to read and understand. Several of the numerical exam ...
Lecture 2
... They are all real (as opposed to imaginary), which can be seen by studying the following (and remembering properties of vector magnitudes) We can also show that eigenvectors associated with distinct eigenvalues of a symmetric matrix are orthogonal We can therefore write a symmetric matrix (I d ...
... They are all real (as opposed to imaginary), which can be seen by studying the following (and remembering properties of vector magnitudes) We can also show that eigenvectors associated with distinct eigenvalues of a symmetric matrix are orthogonal We can therefore write a symmetric matrix (I d ...
Algebraic methods 1 Introduction 2 Perfect matching in
... Definition 13 (Straight-line program). Given a set S, a straight-line program (SLP) is a family of functions F = {fi : 1 ≤ i ≤ m}, fi : S n+i−1 → S for some fixed n ∈ N. An SLP is evaluated on a tuple (s1 , . . . , sn ) by recursion in the following way: f1 is evaluated on (s1 , . . . , sn ) as a fu ...
... Definition 13 (Straight-line program). Given a set S, a straight-line program (SLP) is a family of functions F = {fi : 1 ≤ i ≤ m}, fi : S n+i−1 → S for some fixed n ∈ N. An SLP is evaluated on a tuple (s1 , . . . , sn ) by recursion in the following way: f1 is evaluated on (s1 , . . . , sn ) as a fu ...
On the second dominant eigenvalue affecting the power method for
... Our interest is motivated by recent applications of the power method to compute the limiting probability distribution vector associated with a Markov chain with memory [8]. Two questions arise and we intend to offer a clarification in this presentation. First, the assumption used to propose the Z-ei ...
... Our interest is motivated by recent applications of the power method to compute the limiting probability distribution vector associated with a Markov chain with memory [8]. Two questions arise and we intend to offer a clarification in this presentation. First, the assumption used to propose the Z-ei ...
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.