A Toeplitz matrix is persymmetric, so is its inverse.

A Toeplitz matrix is persymmetric, so is its inverse.
http://www.math.purdue.edu/~xiaj/teaching/692.11s/notes4.pdf

Persymmetric matrix:

In mathematics, persymmetric matrix may refer to:

1. a square matrix which is symmetric in the northeast-to-southwest diagonal; or
2. a square matrix such that the values on each line perpendicular to the main diagonal are the same for a given line.

http://en.wikipedia.org/wiki/Persymmetric_matrix

Linear Algebra Preliminaries and Set Theory

http://www.egr.msu.edu/classes/me851/jchoi/lecture/Linear_Algebra_Notes_Packard.pdf

Quick Description of SVD

http://web.eecs.utk.edu/~dongarra/etemplates/node40.html

Matrix2latex (Matlab)

Little Script for Matrix in Matlab output to Latex
http://www.mathworks.de/matlabcentral/fileexchange/4894-matrix2latex

Fredholm Integral Equation of the First Kind

Equation of the first kind

Integral equations, most generally, are common and take many specific forms (Fourier, Laplace, Hankel, etc.). They each differ in their kernels (defined below). What is distinctive about Fredholm integral equations is that they are integral equations in which the integration limits are constants (they do not include the variable). This is contrast to Volterra integral equations.

A homogeneous Fredholm equation of the first kind is written as:

General theory

The general theory underlying the Fredholm equations is known as Fredholm theory. One of the principal results is that the kernel K is a compact operator, known as the Fredholm operator. Compactness may be shown by invoking equicontinuity. As an operator, it has a spectral theory that can be understood in terms of a discrete spectrum of eigenvalues that tend to 0.

[edit]Applications

Fredholm equations arise naturally in the theory of signal processing, most notably as the famous spectral concentration problem popularized by David Slepian. They also commonly arise in linear forward modeling and inverse problems.

Perturbative inversion methods

Perturbative inversion methods for obtaining bottom geoacoustic parameters

Attention is focused primarily on the modal eigenvalue inverse problem for which the theory for determining the compressional  wave speed, compressional wave attenuation, and density is developed in detail. Properties of this technique are studied using synthetic data and include investigations of the dependence of the results on acoustics frequency, number of modes excited, and partial a priori knowledge of the bottom.

Inversion Problem:
The required input data are trapped mode eigenvalues for one or more frequencies, the group velocity dispersion curves for one or more modes, or the cw pressure field versus range (complex field or magnitude only) .

A great explanation of inverse problem:
An inverse problem is a general framework that is used to convert observed measurements into information about a physical object or system that we are interested in. For example, if we have measurements of the Earth's gravity field, then we might ask the question: "given the data that we have available, what can we say about the density distribution of the Earth in that area?" The solution to this problem (i.e. the density distribution that best matches the data) is useful because it generally tells us something about a physical parameter that we cannot directly observe. Thus, inverse problems are one of the most important, and well-studied mathematical problems in science and mathematics. Inverse problems arise in many branches of science and mathematics, including: computer vision, machine learning, statistics, statistical inference, geophysics, medical imaging (such as computed axial tomographyand EEG/ERP), remote sensing, ocean acoustic tomography, nondestructive testing, astronomy, physics and many other fields.


(Linear Inverse Theory)

The objective of an inverse problem is to find the best model, m, such that (at least approximately)

where G is an operator describing the explicit relationship between the observed data, d, and the model parameters. In various contexts, the operator G is called forward operator,observation operator, or observation function. In the most general context, G represents the governing equations that relate the model parameters to the observed data (i.e. the governing physics).

Mathematical

One central example of a linear inverse problem is provided by a Fredholm first kind integral equation.

Check Fredholm first kind integral equation in next several posts
For sufficiently smooth g the operator defined above is compact on reasonable Banach spaces such as L^p spaces. Even if the mapping is injective its inverse will not be continuous. (However, by the bounded inverse theorem, if the mapping is bijective, then the inverse will be bounded (i.e. continuous).) Thus small errors in the data d are greatly amplified in the solution m. In this sense the inverse problem of inferring m from measured d is ill-posed.


Numerical Scheme (quadrature scheme):
Simpson's Rule
http://en.wikipedia.org/wiki/Simpson's_rule
In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation:





2 Regularization Method

Regularization:
http://en.wikipedia.org/wiki/Regularization_(mathematics)
Regularization of Inverse Problem
http://www.springer.com/mathematics/computational+science+%26+engineering/book/978-0-7923-4157-4

The mathematical term well-posed problem stems from a definition given by Jacques Hadamard. He believed that mathematical models of physical phenomena should have the properties that

1. A solution exists
2. The solution is unique
3. The solution depends continuously on the data, in some reasonable topology.

http://en.wikipedia.org/wiki/Ill-posed_problem

In mathematics and statistics, particularly in the fields of machine learning and inverse problems, regularization involves introducing additional information in order to solve an ill-posed problem or to prevent overfitting. This information is usually of the form of a penalty for complexity, such as restrictions for smoothness or bounds on the vector space norm.

Problems that are not well-posed in the sense of Hadamard are termed ill-posed. Inverse problems are often ill-posed.

Such continuum problems must often be discretized in order to obtain a numerical solution. While in terms of functional analysis such problems are typically continuous, they may suffer from numerical instability when solved with finite precision, or with errors in the data. Even if a problem is well-posed, it may still be ill-conditioned, meaning that a small error in the initial data can result in much larger errors in the answers. An ill-conditioned problem is indicated by a large condition number.

If the problem is well-posed, then it stands a good chance of solution on a computer using a stable algorithm. If it is not well-posed, it needs to be re-formulated for numerical treatment. Typically this involves including additional assumptions, such as smoothness of solution. This process is known as regularization and Tikhonov regularization is one of the most commonly used for regularization of linear ill-posed problems.

Other Keywords:
Normed Vector Space/Vector Space
minimum norm solution
A MatLab Program: Least squires with solving minimum norm solution
http://www.mathworks.com/matlabcentral/fileexchange/17474


Smooth Function:
In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. Higher order differentiability classes correspond to the existence of more derivatives. Functions that have derivatives of all orders are called smooth.

Convex Optimization

http://web.njit.edu/~simeone/ece788.htm

Fundamental theorem of linear algebra

In mathematics, the fundamental theorem of linear algebra makes several statements regarding vector spaces. These may be stated concretely in terms of the rank r of an m×nmatrix A and its singular value decomposition:

First, each matrix  (A has m rows and n columns) induces four fundamental subspaces. These fundamental subspaces are:

name of subspace	definition	containing space	dimension	basis
column space, range or image	im(A) or range(A)		r (rank)	The first r columns of
nullspace or kernel	ker(A) or null(A)		n − r (nullity)	The last (n − r) columns of
row space or coimage	im(AT) or range(AT)		r	The first r rows of
left nullspace or cokernel	ker(AT) or null(AT)		m − r	The last (m − r) rows of

Range of a matrix (Column Space)

In linear algebra, the column space of a matrix (sometimes called the range of a matrix) is the set of all possible linear combinations of its column vectors. The column space of an m × n matrix is a subspace of m-dimensional Euclidean space. The dimension of the column space is called the rank of the matrix.


The column space of a matrix is the image or range of the corresponding matrix transformation.
The dimension of the column space is called the rank of the matrix.


Definition:

Let A be an m × n matrix, with column vectors v₁, v₂, ..., v_n. A linear combination of these vectors is any vector of the form

where c₁, c₂, ..., c_n are scalars. The set of all possible linear combinations of v₁,...,v_n is called the column space of A. That is, the column space of A is the span of the vectorsv₁,...,v_n.

Basis

The columns of A span the column space, but they may not form a basis if the column vectors are not linearly independent. Fortunately, elementary row operations do not affect the dependence relations between the column vectors. This makes it possible to use row reduction to find a basis for the column space.

The left null space of A is the set of all vectors x such that x^TA = 0^T. It is the same as the null space of the transpose of A. The left null space is the orthogonal complement to the column space of A.

This can be seen by writing the product of the matrix A^T and the vector x in terms of the dot product of vectors:

Subgradient Method

From wikipedia
Subgradient methods are iterative methods for solving convex minimization problems. Originally developed by Naum Z. Shor and others in the 1960s and 1970s, subgradient methods are convergent when applied even to a non-differentiable objective function. When the objective function is differentiable, subgradient methods for unconstrained problems use the same search direction as the method of steepest descent.
Subgradient projection methods are often applied to large-scale problems with decomposition techniques. Such decomposition methods often allow a simple distributed method for a problem.


http://en.wikipedia.org/wiki/Subgradient_method

Semi-definite and Gramian matrix

It is called positive-semidefinite (or sometimes nonnegative-definite) if

A matrix M is positive-semidefinite if and only if it arises as the Gram matrix of some set of vectors. In contrast to the positive-definite case, these vectors need not be linearly independent.

Gram matrix:

In linear algebra, the Gramian matrix (or Gram matrix or Gramian) of a set of vectors  in an inner product space is the Hermitian matrix of inner products, whose entries are given by .

http://en.wikipedia.org/wiki/Positive-semidefinite_matrix#Negative-definite.2C_semidefinite_and_indefinite_matrices