Matrix computation for engineers and scientists by Alan Jennings

By Alan Jennings

A e-book for engineers who desire to use matrices in electronic computation. the most subject is matrix numerical research, really the answer of linear simultaneous equations and eigen-value difficulties. chosen purposes were brought and sure positive aspects of machine implementation were mentioned

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16 THE EIGENVALUES OF A MATRIX An eigenvalue and corresponding eigenvector of a matrix satisfy the property that the eigenvector multiplied by the matrix yields a vector proportional to itself. The constant of proportionality is known as the eigenvalue. 92) showing that it has an eigenvalue equal to 4 with a corresponding eigenvector {2 1 o}. As the eigenvector has been premultiplied by the matrix it is known as a right eigenvector. 93) For this equation to be conformable A must be square. Hence only square matrices have eigenvalues.

It has been assumed that no two branches connect the same pair of nodes and also that no branch connects a node with itself. 1 and so produce a general program which constructs the node conductance equations for networks of any topology. e. 11) iDE iEA it will be noted that there is a transpose relationship between the matrices of coefficients. 14) i = G(v - vo) where G is a diagonal matrix of branch conductances and Vo is a column vector of applied voltages. 15) in which AT GA is the node conductance matrix and AT Gvo is the right-hand vector.

The above norm takes no account of the magnitude of the smaller elements of the vector. 116) For x = {7 -10 6}, II x liE =:. 6. 117) for which the Euclidean norm corresponds to b = 2 and the norm based on the magnitude of the largest element corresponds to b -+ 00. 122) The condition qJqj = 0 is called the orthogonality condition for the eigenvectors. 124) I Designating the compounded eigenvecto set [ql q2 ... 125) Any real matrix Q satisfying this equation is known as an orthogonal matrix. 125).

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