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shankar
- 17
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can anyone explain the the real meaning and purpose of eigen vlaue and eigen vectors..
Originally posted by Oxymoron
(off-topic) Has this got anything to do with the Kronecker Delta in Tensor Calculus?
Eigenvalues and eigenvectors are concepts in linear algebra that are used to understand linear transformations. An eigenvalue is a scalar value that represents how much a given vector is stretched or compressed when it is multiplied by a transformation matrix. An eigenvector is a vector that, when multiplied by a transformation matrix, results in a scalar multiple of itself, known as the eigenvalue.
To find eigenvalues and eigenvectors, you must first create a square matrix representing the linear transformation. Then, you can use the characteristic equation (det(A - λI) = 0) to find the eigenvalues, where A is the transformation matrix and λ is the eigenvalue. Once the eigenvalues are found, you can use them to solve for the corresponding eigenvectors using the equation (A - λI)x = 0, where x is the eigenvector.
Eigenvalues and eigenvectors are significant because they provide insight into how a linear transformation affects a given vector. They can also be used to simplify complex calculations and make it easier to understand the behavior of a system. Additionally, they are used in many real-world applications, such as image processing and data compression.
Yes, a matrix can have multiple eigenvalues and corresponding eigenvectors. In fact, for an n x n matrix, there can be up to n distinct eigenvalues. This means that there can also be multiple eigenvectors associated with each eigenvalue.
Eigenvalues and eigenvectors are commonly used in data analysis to reduce the dimensionality of a dataset. This means that instead of representing data with a large number of variables, it can be represented with a smaller set of variables that capture the most important information. This is known as principal component analysis and is often used in fields such as machine learning and data mining.