Elementary Row Operations and Their Inverse
... In the previous section, we mentioned that invertible matrices will play an important in our study of linear algebra; thus it will be helpful to be able to 1. determine whether or not a specific matrix has an inverse, and 2. find inverses when they exist. The following theorem will provide us with sev ...
... In the previous section, we mentioned that invertible matrices will play an important in our study of linear algebra; thus it will be helpful to be able to 1. determine whether or not a specific matrix has an inverse, and 2. find inverses when they exist. The following theorem will provide us with sev ...
A SCHUR ALGORITHM FOR COMPUTING MATRIX PTH ROOTS 1
... The plot of the square root case shows that points in the right half plane are iterated to the positive square root of unity and points in the left half plane to −1. The boundary of these two regions is the imaginary axis, which constitutes the set of initial points for which the Newton iteration fa ...
... The plot of the square root case shows that points in the right half plane are iterated to the positive square root of unity and points in the left half plane to −1. The boundary of these two regions is the imaginary axis, which constitutes the set of initial points for which the Newton iteration fa ...
Minimum Polynomials of Linear Transformations
... The reader should also note that throughout the paper, we will often state that a polynomial p(x) is satisfied by T (v), for some appropriate endomorphism T and vector v. This, however, is a misnomer, as T (v) is a vector, and we have not provided a definition for vector multiplication other than fo ...
... The reader should also note that throughout the paper, we will often state that a polynomial p(x) is satisfied by T (v), for some appropriate endomorphism T and vector v. This, however, is a misnomer, as T (v) is a vector, and we have not provided a definition for vector multiplication other than fo ...
MTH6140 Linear Algebra II 1 Vector spaces
... Theorem 1.2 (The Exchange Lemma) Let V be a vector space over K. Suppose that the vectors v1 , . . . , vn are linearly independent, and that the vectors w1 , . . . , wm are linearly independent, where m > n. Then we can find a number i with 1 ≤ i ≤ m such that the vectors v1 , . . . , vn , wi are li ...
... Theorem 1.2 (The Exchange Lemma) Let V be a vector space over K. Suppose that the vectors v1 , . . . , vn are linearly independent, and that the vectors w1 , . . . , wm are linearly independent, where m > n. Then we can find a number i with 1 ≤ i ≤ m such that the vectors v1 , . . . , vn , wi are li ...
handout2 - UMD MATH
... The condition on the sum of the coefficients provides the distinction between the definition of linear independence and the slightly less restrictive definition of general position. The relationship is made more precise in the following proposition. ...
... The condition on the sum of the coefficients provides the distinction between the definition of linear independence and the slightly less restrictive definition of general position. The relationship is made more precise in the following proposition. ...
Some Results on the Multivariate Truncated Normal Distribution
... This note presents a few results for the multivariate truncated normal distribution. The results may be particularly useful in economic applications where truncated random variables are used to describe data generation processes (e.g., see [4]). Results on linear transformations simply that sums of ...
... This note presents a few results for the multivariate truncated normal distribution. The results may be particularly useful in economic applications where truncated random variables are used to describe data generation processes (e.g., see [4]). Results on linear transformations simply that sums of ...
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.