chiro said:Hey cragar.
The topic that deals with this kind of thing is the Hilbert-Space theory that deals with operator algebras in infinite-dimensional spaces.
If you want to look into this look into things like Hilbert-Space Theory, Banach Spaces, and operator algebras like C* algebras as well as functional analysis in the infinite-dimensional spaces.
An eigenvalue is a scalar value that represents the scale factor of a vector when it is multiplied by a matrix. It is important in matrix calculations because it helps to identify special properties of a matrix, such as its stability, invertibility, and diagonalizability.
Finding the eigenvalues of an infinite matrix can be a complex process. One method is to use the power method, where the matrix is repeatedly multiplied by a vector until the vector converges to an eigenvector. Another method is to use diagonalization, where the matrix is transformed into a diagonal matrix and the eigenvalues are read off the diagonal.
The determinant of an infinite matrix is a value that represents the scale factor of the transformation represented by the matrix. It can be calculated using various methods such as cofactor expansion, Gaussian elimination, or by using the properties of determinants, such as the fact that the determinant of a diagonal matrix is the product of its diagonal entries.
Yes, an infinite matrix can have multiple eigenvalues. In fact, most matrices have multiple eigenvalues, unless they are specially constructed to have only one eigenvalue.
The eigenvalues and determinant of an infinite matrix can be used in a variety of practical applications, such as in physics, engineering, and computer graphics. For example, eigenvalues can be used to determine the stability of a system, while the determinant can be used to calculate the volume of a shape in 3D space.