Eigenvalues and eigenvectors and Projection (linear algebra)

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Difference between Eigenvalues and eigenvectors and Projection (linear algebra)

Eigenvalues and eigenvectors vs. Projection (linear algebra)

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself such that.

Banach space

In mathematics, more specifically in functional analysis, a Banach space (pronounced) is a complete normed vector space.

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Bounded operator

In functional analysis, a bounded linear operator is a linear transformation L between normed vector spaces X and Y for which the ratio of the norm of L(v) to that of v is bounded above by the same number, over all non-zero vectors v in X. In other words, there exists some M\ge 0 such that for all v in X The smallest such M is called the operator norm \|L\|_ \, of L. A bounded linear operator is generally not a bounded function; the latter would require that the norm of L(v) be bounded for all v, which is not possible unless L(v).

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Characteristic polynomial

In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots.

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Conjugate transpose

In mathematics, the conjugate transpose or Hermitian transpose of an m-by-n matrix A with complex entries is the n-by-m matrix A∗ obtained from A by taking the transpose and then taking the complex conjugate of each entry.

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Diagonalizable matrix

In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P−1AP is a diagonal matrix.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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Eigenvalue algorithm

In numerical analysis, one of the most important problems is designing efficient and stable algorithms for finding the eigenvalues of a matrix.

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Functional analysis

Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear functions defined on these spaces and respecting these structures in a suitable sense.

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Hilbert space

The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space.

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Householder transformation

In linear algebra, a Householder transformation (also known as a Householder reflection or elementary reflector) is a linear transformation that describes a reflection about a plane or hyperplane containing the origin.

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Invariant subspace

In mathematics, an invariant subspace of a linear mapping T: V → V from some vector space V to itself is a subspace W of V that is preserved by T; that is, T(W) ⊆ W.

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Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.

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Linear algebra

Linear algebra is the branch of mathematics concerning linear equations such as linear functions such as and their representations through matrices and vector spaces.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Linear subspace

In linear algebra and related fields of mathematics, a linear subspace, also known as a vector subspace, or, in the older literature, a linear manifold, is a vector space that is a subset of some other (higher-dimension) vector space.

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Matrix (mathematics)

In mathematics, a matrix (plural: matrices) is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns.

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Matrix multiplication

In mathematics, matrix multiplication or matrix product is a binary operation that produces a matrix from two matrices with entries in a field, or, more generally, in a ring or even a semiring.

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Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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Scalar (mathematics)

A scalar is an element of a field which is used to define a vector space.

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Self-adjoint operator

In mathematics, a self-adjoint operator on a finite-dimensional complex vector space V with inner product \langle\cdot,\cdot\rangle is a linear map A (from V to itself) that is its own adjoint: \langle Av,w\rangle.

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Spectrum (functional analysis)

In mathematics, particularly in functional analysis, the spectrum of a bounded operator is a generalisation of the set of eigenvalues of a matrix.

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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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