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Hadamard Inequality
In mathematics, Hadamard's inequality (also known as Hadamard's theorem on determinants) is a result first published by Jacques Hadamard in 1893.Maz'ya & Shaposhnikova It is a bound on the determinant of a matrix whose entries are complex numbers in terms of the lengths of its column vectors. In geometrical terms, when restricted to real numbers, it bounds the volume in Euclidean space of ''n'' dimensions marked out by ''n'' vectors ''vi'' for 1 ≤ ''i'' ≤ ''n'' in terms of the lengths of these vectors , , ''vi'', , . Specifically, Hadamard's inequality states that if ''N'' is the matrix having columns ''vi'', then :\left, \det(N) \ \le \prod_^n \, v_i\, . If the ''n'' vectors are non-zero, equality in Hadamard's inequality is achieved if and only if the vectors are orthogonal. Alternate forms and corollaries A corollary is that if the entries of an ''n'' by ''n'' matrix ''N'' are bounded by ''B'', so , ''Nij'', ≤ ''B'' for all ''i'' and ''j'', then :\left, \det(N) \ \le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjugate Transpose
In mathematics, the conjugate transpose, also known as the Hermitian transpose, of an m \times n complex matrix \mathbf is an n \times m matrix obtained by transposing \mathbf and applying complex conjugation to each entry (the complex conjugate of a+ib being a-ib, for real numbers a and b). There are several notations, such as \mathbf^\mathrm or \mathbf^*, \mathbf', or (often in physics) \mathbf^. For real matrices, the conjugate transpose is just the transpose, \mathbf^\mathrm = \mathbf^\operatorname. Definition The conjugate transpose of an m \times n matrix \mathbf is formally defined by where the subscript ij denotes the (i,j)-th entry (matrix element), for 1 \le i \le n and 1 \le j \le m, and the overbar denotes a scalar complex conjugate. This definition can also be written as :\mathbf^\mathrm = \left(\overline\right)^\operatorname = \overline where \mathbf^\operatorname denotes the transpose and \overline denotes the matrix with complex conjugated entries. Other na ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Identity Matrix
In linear algebra, the identity matrix of size n is the n\times n square matrix with ones on the main diagonal and zeros elsewhere. It has unique properties, for example when the identity matrix represents a geometric transformation, the object remains unchanged by the transformation. In other contexts, it is analogous to multiplying by the number 1. Terminology and notation The identity matrix is often denoted by I_n, or simply by I if the size is immaterial or can be trivially determined by the context. I_1 = \begin 1 \end ,\ I_2 = \begin 1 & 0 \\ 0 & 1 \end ,\ I_3 = \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end ,\ \dots ,\ I_n = \begin 1 & 0 & 0 & \cdots & 0 \\ 0 & 1 & 0 & \cdots & 0 \\ 0 & 0 & 1 & \cdots & 0 \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ 0 & 0 & 0 & \cdots & 1 \end. The term unit matrix has also been widely used, but the term ''identity matrix'' is now standard. The term ''unit matrix'' is ambiguous, because it is also used for a matrix of on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diagonalizable
In linear algebra, a square matrix A is called diagonalizable or non-defective if it is similar to a diagonal matrix. That is, if there exists an invertible matrix P and a diagonal matrix D such that . This is equivalent to (Such D are not unique.) This property exists for any linear map: for a finite-dimensional vector space a linear map T:V\to V is called diagonalizable if there exists an ordered basis of V consisting of eigenvectors of T. These definitions are equivalent: if T has a matrix representation A = PDP^ as above, then the column vectors of P form a basis consisting of eigenvectors of and the diagonal entries of D are the corresponding eigenvalues of with respect to this eigenvector basis, T is represented by Diagonalization is the process of finding the above P and and makes many subsequent computations easier. One can raise a diagonal matrix D to a power by simply raising the diagonal entries to that power. The determinant of a diagonal matrix is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermitian Matrix
In mathematics, a Hermitian matrix (or self-adjoint matrix) is a complex square matrix that is equal to its own conjugate transpose—that is, the element in the -th row and -th column is equal to the complex conjugate of the element in the -th row and -th column, for all indices and : A \text \quad \iff \quad a_ = \overline or in matrix form: A \text \quad \iff \quad A = \overline . Hermitian matrices can be understood as the complex extension of real symmetric matrices. If the conjugate transpose of a matrix A is denoted by A^\mathsf, then the Hermitian property can be written concisely as A \text \quad \iff \quad A = A^\mathsf Hermitian matrices are named after Charles Hermite, who demonstrated in 1855 that matrices of this form share a property with real symmetric matrices of always having real eigenvalues. Other, equivalent notations in common use are A^\mathsf = A^\dagger = A^\ast, although in quantum mechanics, A^\ast typically means the complex conjugate onl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inequality Of Arithmetic And Geometric Means
Inequality may refer to: * Inequality (mathematics), a relation between two quantities when they are different. * Economic inequality, difference in economic well-being between population groups ** Income inequality, an unequal distribution of income ** Wealth inequality, an unequal distribution of wealth ** Spatial inequality, the unequal distribution of income and resources across geographical regions ** International inequality, economic differences between countries * Social inequality, unequal opportunities and rewards for different social positions or statuses within a group ** Gender inequality, unequal treatment or perceptions due to gender ** Racial inequality, social distinctions between racial and ethnic groups within a society * Health inequality, differences in the quality of health and healthcare across populations * Educational inequality, the unequal distribution of academic resources * Environmental inequality, unequal environmental harms between different neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace (linear Algebra)
In linear algebra, the trace of a square matrix , denoted , is the sum of the elements on its main diagonal, a_ + a_ + \dots + a_. It is only defined for a square matrix (). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, for any matrices and of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the row and column of . The entries of can be real numbers, complex numbers, or more generally elements of a field . The trace is not defined for non-square matrices. Example Let be a matrix, with \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eigenvalue
In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a constant factor \lambda when the linear transformation is applied to it: T\mathbf v=\lambda \mathbf v. The corresponding eigenvalue, characteristic value, or characteristic root is the multiplying factor \lambda (possibly a negative or complex number). Geometrically, vectors are multi-dimensional quantities with magnitude and direction, often pictured as arrows. A linear transformation rotates, stretches, or shears the vectors upon which it acts. A linear transformation's eigenvectors are those vectors that are only stretched or shrunk, with neither rotation nor shear. The corresponding eigenvalue is the factor by which an eigenvector is stretched or shrunk. If the eigenvalue is negative, the eigenvector's direction is reversed. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Proof
A mathematical proof is a deductive reasoning, deductive Argument-deduction-proof distinctions, argument for a Proposition, mathematical statement, showing that the stated assumptions logically guarantee the conclusion. The argument may use other previously established statements, such as theorems; but every proof can, in principle, be constructed using only certain basic or original assumptions known as axioms, along with the accepted rules of inference. Proofs are examples of exhaustive deductive reasoning that establish logical certainty, to be distinguished from empirical evidence, empirical arguments or non-exhaustive inductive reasoning that establish "reasonable expectation". Presenting many cases in which the statement holds is not enough for a proof, which must demonstrate that the statement is true in ''all'' possible cases. A proposition that has not been proved but is believed to be true is known as a conjecture, or a hypothesis if frequently used as an assumption for ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Set
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The image of the standard basis under a rotation or reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via normalization. The choice of an origin and an orthonormal basis forms a coordinate frame known as an ''orthonormal frame''. For a general inner product space V, an orthonormal basis can be used to define normalized orthogonal coordinates on V. Under these coordinates, the inner product becomes a dot product of vectors. Thus the presence of an orthonormal basis reduces th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vector
In mathematics, a unit vector in a normed vector space is a Vector (mathematics and physics), vector (often a vector (geometry), spatial vector) of Norm (mathematics), length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac=(\frac, \frac, ... , \frac) where ‖u‖ is the Norm (mathematics), norm (or length) of u and \, \mathbf\, = (u_1, u_2, ..., u_n). The proof is the following: \, \mathbf\, =\sqrt=\sqrt=\sqrt=1 A unit vector is often used to represent direction (geometry), directions, such as normal directions. Unit vectors are often chosen to form the basis (linear algebra), basis of a vector space, and every vector in the space may be written as a linear combination form of unit vectors. Orthogonal coordinates ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linearly Independent
In the theory of vector spaces, a set of vectors is said to be if there exists no nontrivial linear combination of the vectors that equals the zero vector. If such a linear combination exists, then the vectors are said to be . These concepts are central to the definition of dimension. A vector space can be of finite dimension or infinite dimension depending on the maximum number of linearly independent vectors. The definition of linear dependence and the ability to determine whether a subset of vectors in a vector space is linearly dependent are central to determining the dimension of a vector space. Definition A sequence of vectors \mathbf_1, \mathbf_2, \dots, \mathbf_k from a vector space is said to be ''linearly dependent'', if there exist scalars a_1, a_2, \dots, a_k, not all zero, such that :a_1\mathbf_1 + a_2\mathbf_2 + \cdots + a_k\mathbf_k = \mathbf, where \mathbf denotes the zero vector. This implies that at least one of the scalars is nonzero, say a_1\ne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
