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meteor
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What's the utility of the eigenvectors of a matrix?
I know that is something about quantum mechanics
I know that is something about quantum mechanics
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Originally posted by meteor
What's the utility of the eigenvectors of a matrix?
I know that is something about quantum mechanics
Originally posted by rdt2
Now think of a 3x3 matrix as representing something more complicated than a vector (it's called a tensor but that doesn't matter here). In general, the matrix will have 9 components (in 3-D), three of which are on the 'main diagonal' (top left to bottom right) and the other six of which are not. Again, there is always a coordinate system in which the 6 'off-diagonal' components are zero. In this system, the three components on the diagonal are the eigenvalues.
Originally posted by dg
This is not true! Not all matrices are diagonalizable!
A restriction to symmetric matrices would be more appropriate here as an illustration of the physical properties of eigenvalues...
Eigenvectors are special vectors that do not change direction when multiplied by a matrix. They are important because they can help us understand the behavior of a matrix and its corresponding transformation.
Eigenvectors are used in data analysis to reduce the dimensionality of a dataset and extract important features. They can also be used for clustering and classification tasks.
The eigenvalues associated with eigenvectors represent the amount of stretching or shrinking that occurs in the direction of the eigenvector when the matrix is multiplied by it. They can also indicate the variance or importance of the corresponding eigenvector in the transformation.
Yes, eigenvectors can have negative values. The sign of an eigenvector does not affect its direction or importance in the transformation.
Eigenvectors are linearly independent, meaning they are not linear combinations of each other. This makes them useful for solving systems of linear equations and understanding the linear independence of a matrix.