Year: 2022 | Month: June | Volume 8 | Issue 1
An Analysis the Numerical and Differential Equation
DOI:10.30954/2454-4132.1.2022.2
Abstract:
Remember that, in general, the word scalar is not restricted to real numbers. We are only using real numbers as scalars in this book, but eigenvalues are often complex numbers. Consider the square matrix A. We say that A is an eigen-value of A if there exists a non-zero vector x such that Ax = \x. In this case, x is called an eigen-vector (corresponding to A), and the pair (A,x) is called an eigen-pair for A. Therefore, A and x are an eigenvalue and an eigenvector, respectively, for A. Now that we have seen an eigen-value and an eigen-vector, let’s talk a little more about them. Why did we require that an eigenvector not be zero? If the eigen-vector was zero, the equation Ax = Xx would yield 0 = 0. Since, this equation is always true, it is not an interesting case. Therefore, we define an eigen-vector to be a non-zero vector that satisfies Ax = Xx. However, as we showed in the previous example, an eigen-value can be zero without causing a problem. We usually say that x is an eigen-vector corresponding to the eigen-value A if they satisfy Ax = Xx. Since, each eigen-vector is associated with an eigen-value, we often refer to an x and A that correspond to one another as an eigen-pair. Did we notice that we called x “an” eigen-vector rather than “the” eigen-vector corresponding to A.
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