Glossary

1.4 Inner Product Spaces and Orthogonality

3 min read•july 22, 2024

Inner product spaces are fundamental in mathematical physics, combining vector spaces with a special function that measures angles and lengths. They provide a framework for understanding geometric relationships in abstract spaces, crucial for quantum mechanics and relativity.

and orthonormal bases build on inner product spaces, introducing perpendicular vectors and special coordinate systems. These concepts are essential for simplifying complex problems, decomposing vectors, and analyzing operators in quantum mechanics and other physical theories.

Inner Product Spaces

Properties of inner product spaces

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  • combines a vector space VV over a field FF (real or complex numbers) with an inner product function ⟨⋅,⋅⟩\langle \cdot, \cdot \rangle that maps pairs of vectors to a scalar value in FF
  • Inner product satisfies conjugate symmetry swaps the order of the vectors and takes the complex conjugate of the result
  • Inner product is linear in the second argument can distribute over vector addition and scalar multiplication
  • Inner product is positive definite always non-negative and only zero when the vector is the zero vector

Calculations in inner product spaces

  • Compute inner products by summing the products of corresponding components for real vector spaces or the products of the complex conjugate of the first vector's components with the second vector's components for complex vector spaces
  • Calculate the of a vector ∥v∥\|v\| by taking the square root of the inner product of the vector with itself represents the length or magnitude of the vector
  • Norm satisfies non-negativity, definiteness, absolute homogeneity, and the triangle inequality
  • Compute the distance between two vectors d(u,v)d(u, v) by taking the norm of the difference between the vectors

Orthogonality and Orthonormal Bases

Orthogonality in vector spaces

  • Two vectors are orthogonal if their inner product is zero (perpendicular in Euclidean spaces)
  • An orthogonal set of vectors has pairwise orthogonal vectors
  • An of vectors is an orthogonal set where each vector has a norm of 1 (unit vectors)
  • An orthogonal basis is a basis consisting of an orthogonal set of vectors
  • An orthonormal basis is a basis consisting of an orthonormal set of vectors

Gram-Schmidt orthonormalization process

  1. Start with a linearly independent set of vectors {u1,u2,…,un}\{u_1, u_2, \ldots, u_n\}
  2. Normalize the first vector v1v_1 by dividing it by its norm ∥u1∥\|u_1\|
  3. For each subsequent vector uiu_i, subtract the projections of uiu_i onto the previous orthonormal vectors vjv_j and normalize the result to obtain viv_i
  • The resulting set {v1,v2,…,vn}\{v_1, v_2, \ldots, v_n\} forms an orthonormal basis for the subspace spanned by the original linearly independent set

Adjoint operators and orthogonality

  • The adjoint operator T∗T^* of a linear operator TT on an inner product space satisfies ⟨Tv,u⟩=⟨v,T∗u⟩\langle Tv, u \rangle = \langle v, T^*u \rangle for all vectors uu and vv
  • Adjoint operators have properties such as involutivity, , and reversing the order of composition
  • A self-adjoint (Hermitian) operator equals its own adjoint T=T∗T = T^*
  • Self-adjoint operators have real eigenvalues and orthogonal eigenvectors corresponding to distinct eigenvalues

Key Terms to Review (15)

Complex inner product: A complex inner product is a mathematical operation that combines two vectors in a complex vector space to produce a complex scalar. This operation not only measures the length of the vectors but also quantifies the angle between them, exhibiting properties such as linearity in one argument, conjugate symmetry, and positive definiteness. These features are essential for defining geometric concepts like orthogonality and distance in complex spaces.
Dot Product: The dot product is an algebraic operation that takes two equal-length sequences of numbers, usually coordinate vectors, and returns a single number. This operation helps in finding the angle between vectors and is vital in understanding inner product spaces, orthogonality, and projections. The dot product also reveals important geometric relationships, such as whether two vectors are orthogonal, which is a key feature in vector spaces.
Euclidean Space: Euclidean space refers to a mathematical construct that describes a flat, infinite space characterized by the familiar geometric properties established by Euclid. It serves as the fundamental setting for geometry, where points, lines, and shapes are defined, and extends into various dimensions, allowing for a rigorous exploration of concepts like distance and angles. This notion is crucial for understanding vector spaces, transformations, and inner products, as it provides the groundwork upon which these concepts are built.
Hilbert Space: A Hilbert space is a complete inner product space that provides the framework for much of modern mathematical physics, particularly in quantum mechanics. It extends the concept of finite-dimensional Euclidean spaces to infinite dimensions and supports the geometric interpretation of quantum states and their evolution. This makes Hilbert spaces essential for understanding both classical vector spaces and orthogonality concepts in quantum theory, as well as specific applications like the quantum harmonic oscillator.
Inner Product Space: An inner product space is a vector space equipped with an inner product, which is a mathematical operation that combines two vectors to produce a scalar. This inner product has properties like linearity, symmetry, and positive definiteness, allowing us to define concepts like length and angle between vectors. The structure of an inner product space is crucial for understanding orthogonality and geometric interpretations in higher-dimensional spaces.
Least Squares Approximation: Least squares approximation is a mathematical method used to find the best-fitting curve or line to a set of data points by minimizing the sum of the squares of the differences (the residuals) between the observed values and the values predicted by the model. This method is closely connected to concepts of inner product spaces and orthogonality, where the goal is to minimize the distance in an inner product space between a given point and a subspace spanned by a set of basis vectors, leading to an optimal solution that is orthogonal to the error vector.
Linearity: Linearity refers to a property of mathematical functions where the output is directly proportional to the input, adhering to the principles of superposition. In various mathematical contexts, linearity implies that operations can be performed independently, meaning the sum of the outputs for multiple inputs equals the output for the sum of those inputs. This concept is crucial for understanding inner product spaces, signal processing through Fourier transforms, and maintaining consistent behavior across transformations.
Norm: A norm is a function that assigns a non-negative length or size to each vector in a vector space, providing a way to measure the 'distance' from the origin in terms of the vector's magnitude. Norms are essential in inner product spaces, where they relate directly to concepts like orthogonality and distance between vectors, helping to understand geometric and algebraic properties in these spaces.
Orthogonal Complement: The orthogonal complement of a subset of a vector space consists of all vectors that are orthogonal to every vector in that subset. This concept is crucial in understanding inner product spaces, as it helps define relationships between subspaces, enabling the study of projections and dimensions within those spaces.
Orthogonal Projection: Orthogonal projection is a method of projecting a vector onto a subspace such that the line connecting the original vector and its projection is perpendicular to that subspace. This concept is central to understanding inner product spaces and orthogonality, as it highlights how vectors can be represented in relation to one another, allowing for decomposition into components that are parallel and orthogonal to given subspaces.
Orthogonality: Orthogonality is a concept that describes the perpendicularity of vectors or functions in a given space, meaning that their inner product is zero. This property is crucial for various mathematical and physical applications, allowing different functions or vectors to maintain independence from one another. It plays a significant role in simplifying complex problems, facilitating analysis in different coordinate systems, and optimizing solutions in series expansions and special functions.
Orthonormal Set: An orthonormal set is a collection of vectors in an inner product space that are both orthogonal and normalized. This means that each pair of distinct vectors in the set is orthogonal to one another, and each vector has a unit length (or norm). Orthonormal sets are essential in many mathematical and physical applications, as they provide a convenient basis for expressing vectors and simplifying calculations in inner product spaces.
Positivity: Positivity refers to a fundamental property of inner products in vector spaces, ensuring that the inner product of any vector with itself is non-negative. This characteristic is crucial because it establishes a notion of length or magnitude, allowing us to measure distances and angles within the space. Positivity also plays a significant role in determining orthogonality, as it ensures that distinct vectors can be compared meaningfully based on their inner products.
Pythagorean Theorem in Inner Product Spaces: The Pythagorean theorem in inner product spaces states that for any two orthogonal vectors, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. This extends the classical Pythagorean theorem from Euclidean geometry into more abstract vector spaces, showcasing how the concept of orthogonality plays a crucial role in determining distances and angles between vectors in inner product spaces.
Riesz Representation Theorem: The Riesz Representation Theorem is a fundamental result in functional analysis that establishes a relationship between continuous linear functionals and inner product spaces. This theorem states that for every continuous linear functional on a Hilbert space, there exists a unique vector in that space such that the functional can be represented as an inner product with that vector. This connection emphasizes the role of inner products in defining and understanding linear functionals in Hilbert spaces.
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