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mind0nmath
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How do i find the eigenvalues and eigenvectors for the linear operator T defined as
T(w,z) = (z,w)??
T(w,z) = (z,w)??
mind0nmath said:How do i find the eigenvalues and eigenvectors for the linear operator T defined as
T(w,z) = (z,w)??
One of the things you should have learned long ago is that you approach problems like this by looking at simple cases: if n= 2, this says T(x,y)= (x+ y, x+ y). In particular, T(1, 0)= (1, 1) and T(0,1)= (1, 1). Yes, the columns of the matrix representing this linear operator in the standard basis are all 1s. The matrix representing this linear operator in the standard basis consists of all 1s.mind0nmath said:how about for something like: T(x_1,x_2,...,x_n) = (x_1+x_2+...+x_n, x_1+x_2+...+x_n, ..., x_1+x_2+...+x_n). The matrix with respect to standard basis would have 1's everywhere? any clues to finding the eigenvalues/vectors?
Eigenvalues and eigenvectors are concepts in linear algebra that are used to represent the characteristics of a matrix. Eigenvalues are scalar values that represent the amount by which an eigenvector is scaled when multiplied by a matrix. Eigenvectors are non-zero vectors that, when multiplied by a matrix, result in a scalar multiple of the original vector.
Eigenvalues and eigenvectors are important because they are used to solve many problems in linear algebra, such as finding the roots of a polynomial or solving systems of differential equations. They also have applications in fields such as physics, engineering, and computer science.
To find eigenvalues and eigenvectors, you first need to find the characteristic polynomial of the matrix. This is done by subtracting the variable λ from the diagonal elements of the matrix and finding the determinant. The eigenvalues are then the values of λ that make the determinant equal to 0. Once you have the eigenvalues, you can find the eigenvectors by solving a system of equations using the eigenvalues.
No, not all matrices have eigenvalues and eigenvectors. Only square matrices (matrices with the same number of rows and columns) have eigenvalues and eigenvectors. Additionally, not all square matrices have distinct eigenvalues, which means they may have repeated or complex eigenvalues.
In data analysis, eigenvalues and eigenvectors are used in techniques such as principal component analysis (PCA) and singular value decomposition (SVD). These methods use the eigenvalues and eigenvectors of a matrix to reduce the dimensionality of data and identify patterns and relationships between variables.