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This chapter contains the basic methods for solving matrix equations of the form Ax=d where A is an matrix and x and d are column vectors. It assumes most of the algebraic systems to be square so that the number rows (equations) and columns (variables) are equal. Vectors may either be row or column vectors and usually are an ordered list of real or complex numbers. However, they could have more general components such as a phone book where the components are a triples having a name, address and phone number. Four basic operations with vectors are scalar product, addition, augmentation and transpose. Matrices can be used to store a variety of information, which often requires visualizing or modifications. There are four element-wise or array operations that will be very useful. Most realistic models have many more unknowns, and the resulting matrices are much larger. The general matrix-vector and matrix-matrix products will be defined and their properties will described.

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There are various classes of matrices that have an inverse. Triangular matrices with invertible diagonal blocks have inverses. In this chapter four additional classes will be introduced: Schur complement, nonsingular and LU factors, M-matrix and symmetric positive definite (SPD). It restricts the matrices to symmetric positive definite matrices. Although this restriction may seem a little severe, there are a number of important applications, which include some classes of partial differential equations and some classes of least squares problems. In the case of the SPD matrix, the Schur complement is also SPD and the first pivot must be positive and so no row interchanges are required. The importance of general energy formulations is for iterative methods and for boundary values problems.

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The goal is to solve systems with more equations than unknowns where A is https://www.w3.org/1998/Math/MathML" display="inline"> m × n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math3_1.tif"/> and https://www.w3.org/1998/Math/MathML" display="inline"> m > n . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math3_2.tif"/> The least squares solution(s) of https://www.w3.org/1998/Math/MathML" display="inline"> A x = d https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math3_3.tif"/> are equivalent to finding the solution(s) of the normal equations https://www.w3.org/1998/Math/MathML" display="inline"> A T A x = A T d . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math3_4.tif"/> If https://www.w3.org/1998/Math/MathML" display="inline"> A T A https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math3_5.tif"/> is nonsingular, then there is a unique solution. An application to parameter identification is illustrated in the second section. The third and fourth sections construct multiple solutions. This is done by using the basis for the subspace https://www.w3.org/1998/Math/MathML" display="inline"> R ( A )   https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math3_6.tif"/> and the projection of d onto https://www.w3.org/1998/Math/MathML" display="inline"> R ( A ) . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math3_7.tif"/>

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Elementary matrices will be used to construct the basis for the null space, N(A), and the range space, R(A). This construction illustrates by several examples. The chapter uses row echelon form to find the bases for the range and null spaces. The general case is described in the third section. These bases are not orthonormal, but they still are useful in the general least squares problem. It illustrates how null spaces can be effectively used to solve equilibrium equations that evolve from circuits, structures and fluid flow models. The chapter allows the reader to identify the homogeneous (null) and particular solutions. A variety of applications to more complicated circuits, structures and fluid flows models will be outlined; select one and study it carefully. In order to keep the discussion as simple as possible, consider fluid flow in a plane, and the rate of change of mass for four adjacent squares.

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If A is https://www.w3.org/1998/Math/MathML" display="inline"> m × n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math5_1.tif"/> , then https://www.w3.org/1998/Math/MathML" display="inline"> A : ℝ n → ℝ m . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math5_2.tif"/> Any vector in https://www.w3.org/1998/Math/MathML" display="inline"> ℝ n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math5_3.tif"/> may be written as a sum of vectors in https://www.w3.org/1998/Math/MathML" display="inline"> N ( A ) https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math5_4.tif"/> and in https://www.w3.org/1998/Math/MathML" display="inline"> R ( A T ) . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math5_5.tif"/> Moreover, these two vectors are orthogonal (perpendicular), which is part of the fundamental theorem. The bases for these two subspaces may have orthonormal vectors. This is established by using the factors https://www.w3.org/1998/Math/MathML" display="inline"> A = Q R https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math5_6.tif"/> , where the columns of Q are orthonormal and R is upper triangular.

Pages: 14

Another way to find orthonormal vectors is to consider eigenvectors of a symmetric matrix. An important case is https://www.w3.org/1998/Math/MathML" display="inline"> A T A https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math6_1.tif"/> in the normal equations, and this will form the core the singular value decomposition of an https://www.w3.org/1998/Math/MathML" display="inline"> m × n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math6_2.tif"/> matrix https://www.w3.org/1998/Math/MathML" display="inline"> A = U Σ V T . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math6_3.tif"/> The spectral theorem for real symmetric https://www.w3.org/1998/Math/MathML" display="inline"> n × n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math6_4.tif"/> matrices gives https://www.w3.org/1998/Math/MathML" display="inline"> A = U D U T https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math6_5.tif"/> where D is the diagonal of eigenvalues and the columns of Q are the corresponding eigenvectors.

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In this chapter we consider the more general https://www.w3.org/1998/Math/MathML" display="inline"> m × n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math7_1.tif"/> real matrix A and its factorization https://www.w3.org/1998/Math/MathML" display="inline"> U Σ V T https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math7_2.tif"/> . The factors satisfy the following: U is https://www.w3.org/1998/Math/MathML" display="inline"> m × m https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math7_3.tif"/> , https://www.w3.org/1998/Math/MathML" display="inline"> U T U = I m https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math7_4.tif"/> , V is https://www.w3.org/1998/Math/MathML" display="inline"> n × n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math7_5.tif"/> , https://www.w3.org/1998/Math/MathML" display="inline"> V T V = I n https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math7_6.tif"/> and Σ is https://www.w3.org/1998/Math/MathML" display="inline"> m × n . https://www.w3.org/1999/xlink" xlink:href="https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781003304128/10158a01-d9a6-4f7c-9eed-42d2fddd582c/content/math7_7.tif"/> This is the “full” SVD. First, we establish the “small” version. Then the “full” and “truncated” versions are described.

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Three important applications will be illustrated for image compression, search engines and noise filters. Grayscale images are associated with m×n matrices whose components are integers. In MATLAB one can “view” the image in several ways. First, just inspect the matrix components. Second, use the MATLAB command mesh to generate a surface of the image where the indices of the matrix are on the xy-plane and the intensity of the image is on the z-axis. Third, one can map the matrix into a standard image file such as a.jpg file. The general scheme is to convert the image to a matrix, modify the matrix and then to convert the modified matrix back to a new image. The operation of light/dark corresponds to multiplying the matrix by a constant less/larger than one. The operation of cropping/panorama corresponds to deleting/augmenting rows or columns of the matrix.

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Ill-conditioned matrices have the unpleasant property that small perturbations in the n×n matrix or right side can give large variations to the solution to the unperturbed system. One application is to hazard identification such as a pollutant in a river with point sources along the river. An interesting application is to hazard identification such as a pollutant in a river with point sources along the river. The first numerical experiment uses observation sites 10, 30, 60 and 90 and source sites 20, 35 and 80. The second numerical experiment uses observation sites 10, 30, 60 and 70 and source sites 20, 35 and 80. The third numerical experiment uses observation sites 10 and 30 and source sites 20, 35 and 80.

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Inner products will be generalized so that orthogonal concepts, eigenvectors, eigenvectors, self-adjoint and positive definite operations can be extended. The general inner product of two vector space vectors is satisfied by the inner products of both real and complex vectors. The Schur decomposition of an n×n complex matrix shows that the matrix is unitarily equivalent to an upper triangular matrix. Thus, the eigenvalues of the matrix are the diagonal components of the upper triangular matrix. The boundary conditions can be quite general, but the people assume they are such that the boundary value problem is self-adjoint. The algebraic system may be solved to give an approximation of the weak solution.

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Direct methods such as the many variations of Gauss elimination possibly require large numbers of computations and storage. There are three important alternatives to avoid this difficulty. They involve approximating the matrix by splittings, using conjugate directions and minimizing the residual. The first inverse matrix computation is a generalization of the geometric series to matrices. The second inverse matrix approximation gives a condition on a matrix C so that it will be “near” a nonsingular matrix A and C will also be nonsingular. Iterative methods require testing for possible convergence. The main goal in this chapter is to show the SOR method converges for SPD matrices. The conjugate gradient method has three basic components: steepest descent in multiple directions, conjugate directions and preconditioning. The multiple direction version of steepest descent ensures the largest possible decrease in the “energy”.

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The methods of successive approximations or Picard's method and Newton's method will be studied for systems of equations. In the case of one variable these methods are introduced in calculus as an application of the mean value theorems. The Levenberg-Marquardt method will be used to approximate the solutions to nonlinear least squares problems. Application to parameter identification of differential equations will be illustrated. There are four primary components that govern the growth of an infected population: probability of transmission, the degree of susceptibility, number of contacts with an infected and duration of contacts. This chapter shows the four functions of time can be expressed as function of the cumulated infection, which evolves from a nonlinear differential equation, CI-equation, with the three parameters of the SIRD model. The SIRD model assumes the parameters are constant with respect to the particular population set and the time interval.