A biorthogonal system is a concept in linear algebra and functional analysis, generalizing the idea of an orthogonal or orthonormal basis. It consists of two sets of vectors (or functions) in a vector space, often an inner product space, such that elements from one set are orthogonal to elements from the other set, except for specific paired elements.
Definition
Given a vector space $V$ over a field (typically real or complex numbers) equipped with an inner product $\langle \cdot, \cdot \rangle$, a biorthogonal system consists of two sets of vectors, say ${\phi_i}{i \in I}$ and ${\psi_j}{j \in J}$, where $I$ and $J$ are index sets. These sets are biorthogonal if they satisfy the condition:
$\langle \phi_i, \psi_j \rangle = \delta_{ij}$
where $\delta_{ij}$ is the Kronecker delta, which is 1 if $i=j$ and 0 if $i eq j$.
In simpler terms, each vector $\phi_i$ is orthogonal to every vector $\psi_j$ except when $j=i$, and similarly, each $\psi_i$ is orthogonal to every $\phi_j$ except when $j=i$. The inner product of a vector with its biorthogonal counterpart is typically normalized to 1.
Properties
- Generalization of Orthogonality: An orthonormal basis ${\mathbf{e}_i}$ is a special case of a biorthogonal system where the two sets are identical, i.e., $\phi_i = \psi_i = \mathbf{e}_i$. In this case, $\langle \mathbf{e}_i, \mathbf{e}j \rangle = \delta{ij}$.
- Completeness: In many applications, especially in infinite-dimensional spaces like Hilbert spaces, both sets ${\phi_i}$ and ${\psi_j}$ are complete, meaning that any vector in the space can be represented as a linear combination (or series) of elements from either set.
- Linear Independence: The vectors within each set of a biorthogonal system are linearly independent. If, for instance, $\sum_k c_k \phi_k = 0$, taking the inner product with $\psi_j$ yields $\sum_k c_k \langle \phi_k, \psi_j \rangle = \sum_k c_k \delta_{kj} = c_j = 0$ for all $j$.
- Non-Orthogonality within Sets: Unlike an orthogonal basis, the vectors within the set ${\phi_i}$ are generally not orthogonal to each other (i.e., $\langle \phi_i, \phi_j \rangle eq 0$ for $i eq j$), and similarly for ${\psi_j}$. This allows for more flexibility in constructing such systems.
- Dual Basis: If one set, say ${\phi_i}$, forms a basis for the space, then the other set, ${\psi_j}$, is often referred to as its "dual basis" or "reciprocal basis."
Applications
Biorthogonal systems find widespread use in various fields:
- Quantum Mechanics: In the context of non-Hermitian Hamiltonians or generalized eigenvalue problems, the eigenvectors generally do not form an orthogonal set. However, a biorthogonal system can be constructed between the right-eigenvectors and the left-eigenvectors. This is crucial for defining completeness relations and calculating expectation values.
- Functional Analysis: They are used in the study of generalized Fourier series, particularly when dealing with non-orthogonal bases or frames.
- Signal Processing and Wavelets: Biorthogonal wavelets are a class of wavelets that are designed to have compact support and other desirable properties for signal compression and analysis. The analysis wavelets and synthesis wavelets form a biorthogonal pair. This allows for perfect reconstruction of signals even when the basis functions are not orthogonal to themselves.
- Numerical Analysis: For solving certain types of differential equations or eigenvalue problems, especially those involving non-symmetric matrices, biorthogonal decompositions can provide a stable and efficient framework.
- Computational Fluid Dynamics: Used in the context of spectral methods for fluid simulations where non-orthogonal basis functions might be employed due to boundary conditions.
Construction
Given a basis ${\phi_i}$ for a finite-dimensional space $V$, one can construct its dual basis ${\psi_j}$. If the Gram matrix $G_{ij} = \langle \phi_i, \phi_j \rangle$ is invertible, then the elements of the dual basis can be found. In infinite-dimensional Hilbert spaces, the existence and construction of biorthogonal systems often rely on the Riesz representation theorem and properties of operators.