Selected Problems — Matrix Algebra Math 2300
... 1. Prove that if A is nonsingular then AT is nonsingular and (AT )−1 = (A−1 )T . Discussion: Lets put into words what are we asked to show in this problem. First, we must show that if a matrix is invertible, then so is its transpose. We must also show that “the inverse of the transpose is the same a ...
... 1. Prove that if A is nonsingular then AT is nonsingular and (AT )−1 = (A−1 )T . Discussion: Lets put into words what are we asked to show in this problem. First, we must show that if a matrix is invertible, then so is its transpose. We must also show that “the inverse of the transpose is the same a ...
Fast multiply, nonzero structure
... Note that all we are really using in these routines is the fact that the underlying matrices are low rank. The rank is a property of the underlying linear transformation, independent of basis; that is, rank(A) = rank(BAC) for any nonsingular B and C. So we can still get a fast matrix multiply for  ...
... Note that all we are really using in these routines is the fact that the underlying matrices are low rank. The rank is a property of the underlying linear transformation, independent of basis; that is, rank(A) = rank(BAC) for any nonsingular B and C. So we can still get a fast matrix multiply for  ...
EQUIVALENT REAL FORMULATIONS FOR SOLVING COMPLEX
... A B −y α For future reference, we denote the matrix associated with the K1 to K4 formulations by K1 to K4 , respectively. In fact, it’s to be said that, for any pair of bases for R2n , there exists a different equivalent real formulation of (1). This not standing, K1-K4 are recognized to be among th ...
... A B −y α For future reference, we denote the matrix associated with the K1 to K4 formulations by K1 to K4 , respectively. In fact, it’s to be said that, for any pair of bases for R2n , there exists a different equivalent real formulation of (1). This not standing, K1-K4 are recognized to be among th ...
M341 Linear Algebra, Spring 2014, Travis Schedler Review Sheet
... Describe the following algorithms: Gaussian and Gauss-Jordan elimination, GramSchmidt orthogonalization. Give the statements of the following theorems: the dimension theorem (relating three of the four: domain, codomain, kernel, and image); the Jordan Normal Form theorem for complex n by n matrices; ...
... Describe the following algorithms: Gaussian and Gauss-Jordan elimination, GramSchmidt orthogonalization. Give the statements of the following theorems: the dimension theorem (relating three of the four: domain, codomain, kernel, and image); the Jordan Normal Form theorem for complex n by n matrices; ...
Math 314H Homework # 2 Due: Monday, April 1 Instructions: Do six
... returns a basis for the null space of A ...
... returns a basis for the null space of A ...
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.