Matrices and Linear Algebra
... Definition 2.2.3. Two linear systems Ax = b and Bx = c are called equivalent if one can be converted to the other by elementary equation operations. It is easy to see that this implies the following Theorem 2.2.4. Two linear systems Ax = b and Bx = c are equivalent if and only if both [A|b] and [B|c ...
... Definition 2.2.3. Two linear systems Ax = b and Bx = c are called equivalent if one can be converted to the other by elementary equation operations. It is easy to see that this implies the following Theorem 2.2.4. Two linear systems Ax = b and Bx = c are equivalent if and only if both [A|b] and [B|c ...
Chapter 2 Systems of Linear Equations and Matrices
... An important aspect of this chapter is WORD PROBLEMS. In this section we will be setting up the equations for problems that we will solve later. Example - An apartment complex is being developed that has one-, two- and three-bedroom units. The developer has decided that there will be a total of 192 ...
... An important aspect of this chapter is WORD PROBLEMS. In this section we will be setting up the equations for problems that we will solve later. Example - An apartment complex is being developed that has one-, two- and three-bedroom units. The developer has decided that there will be a total of 192 ...
MODULE 11 Topics: Hermitian and symmetric matrices Setting: A is
... Theorem: hAx, xi > 0 if and only if all eigenvalues of A are strictly positive. Proof: Suppose that hAx, xi > 0. If λ is a non-positive eigenvalue with eigenvector u then hAu, ui = λhu, ui ≤ 0 contrary to assumption. Hence the eigenvalues must be positive. Conversely, assume that all eigenvalues are ...
... Theorem: hAx, xi > 0 if and only if all eigenvalues of A are strictly positive. Proof: Suppose that hAx, xi > 0. If λ is a non-positive eigenvalue with eigenvector u then hAu, ui = λhu, ui ≤ 0 contrary to assumption. Hence the eigenvalues must be positive. Conversely, assume that all eigenvalues are ...
ON BEST APPROXIMATIONS OF POLYNOMIALS IN
... these are the m coefficients at the m + 1 smallest powers of z, namely 1, . . . , z . We start with conditions so that the values of (2.3) and (2.4) are positive for all given nonzero polynomials g ∈ Pℓ and h ∈ Pm , respectively. Lemma 2.1. Consider the approximation problems (2.3) and (2.4), where ...
... these are the m coefficients at the m + 1 smallest powers of z, namely 1, . . . , z . We start with conditions so that the values of (2.3) and (2.4) are positive for all given nonzero polynomials g ∈ Pℓ and h ∈ Pm , respectively. Lemma 2.1. Consider the approximation problems (2.3) and (2.4), where ...
Chapter 4 Basics of Classical Lie Groups: The Exponential Map, Lie
... as subspaces of R ), and in fact, smooth real manifolds. Such objects are called (real) Lie groups. The real vector spaces sl(n, C), u(n), and su(n) are Lie algebras associated with SL(n, C), U(n), and SU(n). The algebra structure is given by the Lie bracket, which is defined as [A, B] = AB − BA. (2 ...
... as subspaces of R ), and in fact, smooth real manifolds. Such objects are called (real) Lie groups. The real vector spaces sl(n, C), u(n), and su(n) are Lie algebras associated with SL(n, C), U(n), and SU(n). The algebra structure is given by the Lie bracket, which is defined as [A, B] = AB − BA. (2 ...
MATRICES Chapter I: Introduction of Matrices 1.1 Definition 1: 1.2
... Definition 6.2.1: A square matrix A is called diagonalizable if there exists an invertible matrix P such that is a diagonal matrix, the matrix P is said to diagonalizable A. Theorem 6.2.1: If A is a square matrix of order n, then the following are equivalent. (i) (ii) ...
... Definition 6.2.1: A square matrix A is called diagonalizable if there exists an invertible matrix P such that is a diagonal matrix, the matrix P is said to diagonalizable A. Theorem 6.2.1: If A is a square matrix of order n, then the following are equivalent. (i) (ii) ...
Research Article Computing the Square Roots of a Class of
... Theorem 3.6. Let A be a nonsingular Hermitian k-circulant matrix, assume that A has a Hermitian k-circulant square matrix. (i) If n/2 is even, then each M and N in 2.7 admits a square root, respectively. (ii) If n/2 is odd, then A’s reduced form RA ∈ Rn×n in 2.12 has a real square root. Proof. ...
... Theorem 3.6. Let A be a nonsingular Hermitian k-circulant matrix, assume that A has a Hermitian k-circulant square matrix. (i) If n/2 is even, then each M and N in 2.7 admits a square root, respectively. (ii) If n/2 is odd, then A’s reduced form RA ∈ Rn×n in 2.12 has a real square root. Proof. ...
B Linear Algebra: Matrices
... is a symmetric matrix of order n. Such an operation is called a congruent or congruential transformation. It is often found in finite element analysis when changing coordinate bases because such a transformation preserves certain key properties (such as symmetry). Loss of Symmetry. The product of tw ...
... is a symmetric matrix of order n. Such an operation is called a congruent or congruential transformation. It is often found in finite element analysis when changing coordinate bases because such a transformation preserves certain key properties (such as symmetry). Loss of Symmetry. The product of tw ...
Multilinear spectral theory
... L. de Lathauwer, B. de Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1, . . . , RN ) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl., 21 (4), 2000, pp. 1324–1342. Same equations first appeared in the context of rank-1 tensor approximations. Our study differs in that we a ...
... L. de Lathauwer, B. de Moor, and J. Vandewalle, “On the best rank-1 and rank-(R1, . . . , RN ) approximation of higher-order tensors,” SIAM J. Matrix Anal. Appl., 21 (4), 2000, pp. 1324–1342. Same equations first appeared in the context of rank-1 tensor approximations. Our study differs in that we a ...
Cramer–Rao Lower Bound for Constrained Complex Parameters
... important tool in the performance evaluation of estimators which arise frequently in the fields of communications and signal processing. Most problems involving the CRB are formulated in terms of unconstrained real parameters [1]. Two useful developments of the CRB theory have been presented in late ...
... important tool in the performance evaluation of estimators which arise frequently in the fields of communications and signal processing. Most problems involving the CRB are formulated in terms of unconstrained real parameters [1]. Two useful developments of the CRB theory have been presented in late ...
Spectrum of certain tridiagonal matrices when their dimension goes
... 2, we show that AN is similar to a symmetric matrix. In this paper, the goal is to determine spAN , the spectrum of AN , when N goes to infinity, and to use this information to find f (AN ) for an entire real valued function f (see [7]). In particular, we are interested to find the limit of f (AN ), as ...
... 2, we show that AN is similar to a symmetric matrix. In this paper, the goal is to determine spAN , the spectrum of AN , when N goes to infinity, and to use this information to find f (AN ) for an entire real valued function f (see [7]). In particular, we are interested to find the limit of f (AN ), as ...
Lecture 20 - Math Berkeley
... - recall also the “key formula” involving the discriminant, which we obtained by completing the square: 4af (x, y) = (2ax + by)2 − dy 2 1. Discriminant and Representability - let f (x, y) be a binary QF. Recall that we say f represents the integer n if there are integers x, y such that f (x, y) = n. ...
... - recall also the “key formula” involving the discriminant, which we obtained by completing the square: 4af (x, y) = (2ax + by)2 − dy 2 1. Discriminant and Representability - let f (x, y) be a binary QF. Recall that we say f represents the integer n if there are integers x, y such that f (x, y) = n. ...
Section 6.1 - Canton Local
... If a matrix A has m rows and n columns, then A is said to be of order m by n (written m × n). The entry or element in the ith row and jth column is a real number and is denoted by the double-subscript notation aij. We can call aij the (i, j)th entry. ...
... If a matrix A has m rows and n columns, then A is said to be of order m by n (written m × n). The entry or element in the ith row and jth column is a real number and is denoted by the double-subscript notation aij. We can call aij the (i, j)th entry. ...
Fixed Point
... Conversely, it is also possible to show that for every ε > 0 there exists a norm k·kε on Rn such that the induced matrix norm k·kε on Rn×n satisfies kAkε ≤ ρ(A) + ε (this is a rather tedious construction involving real versions of the Jordan normal form of A). Together, these results imply that ther ...
... Conversely, it is also possible to show that for every ε > 0 there exists a norm k·kε on Rn such that the induced matrix norm k·kε on Rn×n satisfies kAkε ≤ ρ(A) + ε (this is a rather tedious construction involving real versions of the Jordan normal form of A). Together, these results imply that ther ...