Linear Algebra Notes - An error has occurred.
... 2. Ax = b has a unique solution x for any b, 3. rref(A) = In , 4. rank(A) = n. Proof. Let A be an n × m matrix with inverse A−1 . Since T (x) = Ax maps Rm → Rn , the inverse transformation T −1 maps Rn → Rm , so A−1 is an m × n matrix. If m > n, then the linear system Ax = 0 has at least one free va ...
... 2. Ax = b has a unique solution x for any b, 3. rref(A) = In , 4. rank(A) = n. Proof. Let A be an n × m matrix with inverse A−1 . Since T (x) = Ax maps Rm → Rn , the inverse transformation T −1 maps Rn → Rm , so A−1 is an m × n matrix. If m > n, then the linear system Ax = 0 has at least one free va ...
Chapter 3 Linear Codes
... code C in Fqn is a q n−k × q k array listing all the cosets of Fqn in which the first row consists of the code C with 0 on the extreme left, and the other rows are the cosets ui + C, each arranged in corresponding order, with the coset leader on the left. Remark. The standard array may be constructe ...
... code C in Fqn is a q n−k × q k array listing all the cosets of Fqn in which the first row consists of the code C with 0 on the extreme left, and the other rows are the cosets ui + C, each arranged in corresponding order, with the coset leader on the left. Remark. The standard array may be constructe ...
ON THE NUMBER OF ZERO-PATTERNS OF A SEQUENCE OF
... i=1 λi gi = 0 exists (λi ∈ F ). Let j be a subscript such that |Sj | is minimal among the Si with λi 6= 0. Substitute uj in the relation. While λj gj (uj ) 6= 0, we have λi gi (uj ) = 0 for all i 6= j, a contradiction. Remark 3.1. The history of combinatorial applications of the “linear algebra boun ...
... i=1 λi gi = 0 exists (λi ∈ F ). Let j be a subscript such that |Sj | is minimal among the Si with λi 6= 0. Substitute uj in the relation. While λj gj (uj ) 6= 0, we have λi gi (uj ) = 0 for all i 6= j, a contradiction. Remark 3.1. The history of combinatorial applications of the “linear algebra boun ...
A Superfast Algorithm for Confluent Rational Tangential
... of parameters. Manipulating directly on these parameters allows us to design efficient fast algorithms. One of the most fundamental matrix problems is that of multiplying a (structured) matrix with a vector. Many fundamental algorithms such as convolution, Fast Fourier Transform, Fast Cosine/Sine Tr ...
... of parameters. Manipulating directly on these parameters allows us to design efficient fast algorithms. One of the most fundamental matrix problems is that of multiplying a (structured) matrix with a vector. Many fundamental algorithms such as convolution, Fast Fourier Transform, Fast Cosine/Sine Tr ...
Stability of the Replicator Equation for a Single
... vector. The next step is to obtain the explicit solution (Theorem 2) for the evolution of the covariance matrix. Substitution of this solution into the dynamics for the mean results in a system of linear differential equations with time varying coefficients. The stability analysis of this system for th ...
... vector. The next step is to obtain the explicit solution (Theorem 2) for the evolution of the covariance matrix. Substitution of this solution into the dynamics for the mean results in a system of linear differential equations with time varying coefficients. The stability analysis of this system for th ...
![MATH 108A HW 6 SOLUTIONS Problem 1. [§3.15] Solution. `⇒` Let](http://s1.studyres.com/store/data/007003373_1-43eb22f09357d6b24c4d016966797722-300x300.png)
17_ the assignment problem
... We now show that the algorithm must terminate after a finite number of steps. (We leave it as Exercise 13 to show that, after performing the initial step, all entries in the reduced matrix are nonnegative.) We will show that the sum of all entries in the matrix decreases by at least 1 whenever step ( ...
... We now show that the algorithm must terminate after a finite number of steps. (We leave it as Exercise 13 to show that, after performing the initial step, all entries in the reduced matrix are nonnegative.) We will show that the sum of all entries in the matrix decreases by at least 1 whenever step ( ...
COMPLETELY RANK-NONINCREASING LINEAR MAPS Don
... Suppose S is a linear subspace of B(H) and φ : S → B(M ) is linear. We are not assuming that S is norm-closed or that φ is bounded. We say that φ is a similarity if there is an invertible operator W such that, for every S ∈ S, φ(S) = W −1 SW. We say that φ is a compression if there is an operator V ...
... Suppose S is a linear subspace of B(H) and φ : S → B(M ) is linear. We are not assuming that S is norm-closed or that φ is bounded. We say that φ is a similarity if there is an invertible operator W such that, for every S ∈ S, φ(S) = W −1 SW. We say that φ is a compression if there is an operator V ...
Chapter 2 Determinants
... Chapter 2 Determinants SECTION A Determinant of a Matrix By the end of this section you will be able to evaluate the determinant of various square matrices understand what is meant by the terms cofactor and minor of a matrix determine the inverse of a square matrix T. Seki was the first person ...
... Chapter 2 Determinants SECTION A Determinant of a Matrix By the end of this section you will be able to evaluate the determinant of various square matrices understand what is meant by the terms cofactor and minor of a matrix determine the inverse of a square matrix T. Seki was the first person ...
2. Systems of Linear Equations, Matrices
... Let us review the steps of Example 2.1.2. We copied the first row, then we took 4/2 times the entries of the first row in order to change the 2 into a 4, and subtracted those multiples from the corresponding entries of the second row. (We express this operation more briefly by saying that we subtracted ...
... Let us review the steps of Example 2.1.2. We copied the first row, then we took 4/2 times the entries of the first row in order to change the 2 into a 4, and subtracted those multiples from the corresponding entries of the second row. (We express this operation more briefly by saying that we subtracted ...
Efficient Dense Gaussian Elimination over the Finite Field with Two
... Perhaps, the most well-known of such decompositions is the row echleon form. Definition 3 Row Echelon Form. An m × n matrix A is in row echelon form if any all-zero rows are grouped together at the bottom of the matrix, and if the leading entry of each non-zero row is located to the right of the lea ...
... Perhaps, the most well-known of such decompositions is the row echleon form. Definition 3 Row Echelon Form. An m × n matrix A is in row echelon form if any all-zero rows are grouped together at the bottom of the matrix, and if the leading entry of each non-zero row is located to the right of the lea ...
Math 211
... the set of ordered n-tuples of real numbers: Rn := {(a1 , . . . , an ) | ai ∈ R, 1 ≤ i ≤ n}. The elements of Rn are called points or vectors. Definition 1.6. A function of several variables is a function of the form f : S → Rm where S ⊆ Rn . Writing f (x) = (f1 (x), . . . , fm (x)), the function fi ...
... the set of ordered n-tuples of real numbers: Rn := {(a1 , . . . , an ) | ai ∈ R, 1 ≤ i ≤ n}. The elements of Rn are called points or vectors. Definition 1.6. A function of several variables is a function of the form f : S → Rm where S ⊆ Rn . Writing f (x) = (f1 (x), . . . , fm (x)), the function fi ...
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.