Introduction to Linear Algebra using MATLAB Tutorial
... MATLAB can be used in two basic modes. In the Command Window, you can use it interactively; you type a command or expression and get an immediate result. You can also write programs, using scripts and functions (both of which are stored in M-files). This document does not describe the programming co ...
... MATLAB can be used in two basic modes. In the Command Window, you can use it interactively; you type a command or expression and get an immediate result. You can also write programs, using scripts and functions (both of which are stored in M-files). This document does not describe the programming co ...
Study Guide - URI Math Department
... Theorem 1.6. Let V be a vector space, and let S1 ⊆ S2 ⊆ V . If s1 is linearly dependent, then S2 is linearly dependent. Also, if S2 is linearly independent, then S1 is linearly independent. Cor 2. Let V be a vector space, and let S1 ⊆ S2 ⊆ V . If S2 is linearly independent, then S1 is linearly indep ...
... Theorem 1.6. Let V be a vector space, and let S1 ⊆ S2 ⊆ V . If s1 is linearly dependent, then S2 is linearly dependent. Also, if S2 is linearly independent, then S1 is linearly independent. Cor 2. Let V be a vector space, and let S1 ⊆ S2 ⊆ V . If S2 is linearly independent, then S1 is linearly indep ...
consistency and efficient solution of the sylvester equation for
... A = A∗ and positive definite. The results discussed in this paragraph are the only ones that have been published for the equations (1.2), as far as we know. The necessary and sufficient conditions on the existence and uniqueness of solutions of equations (1.2) developed in [5, 22, 28] are stated and ...
... A = A∗ and positive definite. The results discussed in this paragraph are the only ones that have been published for the equations (1.2), as far as we know. The necessary and sufficient conditions on the existence and uniqueness of solutions of equations (1.2) developed in [5, 22, 28] are stated and ...
Textbook
... • Scalar quantities are written as unbolded greek letters like α, δ, and λ. • The trace of a square matrix An×n , denoted Tr (A) or Trace(A), is the sum of the diagonal elements of A, n ...
... • Scalar quantities are written as unbolded greek letters like α, δ, and λ. • The trace of a square matrix An×n , denoted Tr (A) or Trace(A), is the sum of the diagonal elements of A, n ...
GLn(R) AS A LIE GROUP Contents 1. Matrix Groups over R, C, and
... Proof. First of all, note that the multiplication of elements in R3 is uv = −u · v + u × v, where · and × are the respective dot and cross products in our standard conception of R3 . So if u ∈ iR + jR + kR, then u2 = −1. Since R commutes with all elements of H, t ∈ R means that qtq −1 = t for q 6= 0 ...
... Proof. First of all, note that the multiplication of elements in R3 is uv = −u · v + u × v, where · and × are the respective dot and cross products in our standard conception of R3 . So if u ∈ iR + jR + kR, then u2 = −1. Since R commutes with all elements of H, t ∈ R means that qtq −1 = t for q 6= 0 ...
M04/01
... More sophisticated tools have largely replaced multiplication tables of groups. However when associativity is dropped many of these tools are no longer available and often examples are constructed by indicating a multiplication table. A generalization of Ward quasigroups is obtained when the operati ...
... More sophisticated tools have largely replaced multiplication tables of groups. However when associativity is dropped many of these tools are no longer available and often examples are constructed by indicating a multiplication table. A generalization of Ward quasigroups is obtained when the operati ...
Arrays - Personal
... is a matrix of size (m by n). Sometimes we say “matrix A has dimension (m by n).” The numbers that make up the array are called the elements of the matrix an, in MATLAB, no distinction is made between elements that are real numbers and complex numbers. In the double subscript notation aij for matrix ...
... is a matrix of size (m by n). Sometimes we say “matrix A has dimension (m by n).” The numbers that make up the array are called the elements of the matrix an, in MATLAB, no distinction is made between elements that are real numbers and complex numbers. In the double subscript notation aij for matrix ...
On a classic example in the nonnegative inverse eigenvalue problem
... real numbers, λ1 > 0 and λi ≤ 0, i = 1, 2, . . . , n. In this case it turns out that the condition s1 (σ) = λ1 + λ2 + . . . + λn ≥ 0 is sufficient for realizability of σ. An elegant proof of this result was given in [2], where it is shown that in this special case the companion matrix of the polynomia ...
... real numbers, λ1 > 0 and λi ≤ 0, i = 1, 2, . . . , n. In this case it turns out that the condition s1 (σ) = λ1 + λ2 + . . . + λn ≥ 0 is sufficient for realizability of σ. An elegant proof of this result was given in [2], where it is shown that in this special case the companion matrix of the polynomia ...
Coding Theory: Linear-Error Correcting Codes 1 Basic Definitions
... Definition 2.3 Denote the multiplicative identity of a field F as 1. Then characteristic of F is the least positive integer p such that 1 added to itself p times is equal to 0. We can prove that this characteristic must be either 0 or a prime number. Theorem 2.2 A finite field F of characteristic p ...
... Definition 2.3 Denote the multiplicative identity of a field F as 1. Then characteristic of F is the least positive integer p such that 1 added to itself p times is equal to 0. We can prove that this characteristic must be either 0 or a prime number. Theorem 2.2 A finite field F of characteristic p ...
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.