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An eigenvalue is a scalar value that represents the scaling factor of an eigenvector when a linear transformation is applied to it. In simpler terms, it is a special number associated with a matrix that describes how the matrix stretches or compresses a vector.
To find eigenvalues of a matrix, you need to solve the characteristic equation det(A - λI) = 0, where A is the given matrix and λ is the eigenvalue. This will give you one or more eigenvalues for the matrix.
A basis is a set of linearly independent vectors that span a vector space. It is the minimal set of vectors needed to represent all other vectors in that space. In other words, any vector in the vector space can be written as a linear combination of the basis vectors.
To find the basis of a matrix, you need to first find the eigenvalues of the matrix. Then, for each eigenvalue, find the corresponding eigenvectors. The set of all eigenvectors will form the basis of the matrix.
Finding eigenvalues and basis is important because it helps us understand the behavior of a matrix and its effect on vectors. It is also used in various applications such as solving systems of differential equations, image compression, and data analysis.
