- #1
frederick
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If A & B are linear operators, and AY=aY & BY=bY, what is the relationship between A & B such that e^A*e^B=e^(A+B)?? --where e^x=1+x+x^2/2+x^3/3!+...+x^n/n!
What am I missing? Consider the matrices:then A and B commute.
A linear operator is a mathematical function that maps one vector space to another, and satisfies two properties: additivity and homogeneity. It can be represented by a square matrix and is commonly used in linear algebra to describe transformations.
Eigenvalues and eigenvectors are a pair of values associated with a linear operator. Eigenvalues represent the scalar values that remain unchanged under the linear transformation, while eigenvectors represent the corresponding non-zero vectors that are only scaled by the linear operator.
To find eigenvalues and eigenvectors, you can solve the characteristic equation det(A-λI) = 0, where A is the square matrix representing the linear operator and λ is the eigenvalue. Once you have the eigenvalues, you can find the corresponding eigenvectors by solving the system of linear equations (A-λI)v = 0, where v is the eigenvector.
Eigenvalues and eigenvectors are important in linear algebra because they provide a way to understand the behavior of a linear operator. They can be used to find the direction and magnitude of transformations, and to identify special properties of the linear operator, such as whether it is invertible or has complex eigenvalues.
Eigenvalues and eigenvectors have many practical applications, including in physics, engineering, and data analysis. They are used to analyze systems that involve linear transformations, such as vibration modes in mechanical systems, electrical circuits, and quantum mechanics. In data analysis, they can be used to reduce the dimensionality of a dataset and identify important patterns or features.