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Eigenvalues
Eigenvalues An eigenvalue is a scalar value associated with a linear transformation that represents a change in direction and magnitude of the vector. In...
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Eigenvalues An eigenvalue is a scalar value associated with a linear transformation that represents a change in direction and magnitude of the vector. In...
An eigenvalue is a scalar value associated with a linear transformation that represents a change in direction and magnitude of the vector. In simpler terms, it tells us how much the vector changes under the transformation, and how this change depends on the direction of the vector.
Let's consider a 2D transformation represented by a matrix:
The eigenvalues of A are the roots of the characteristic polynomial, which is a polynomial defined as:
where:
A is the transformation matrix.
\lambda is a scalar (eigenvalue).
I is the identity matrix.
The characteristic polynomial tells us about the number and nature of the eigenvalues of A.
Positive eigenvalues correspond to stretching the vectors along the corresponding direction.
Negative eigenvalues correspond to contracting the vectors along the corresponding direction.
Zero eigenvalues correspond to vectors being preserved by the transformation.
The eigenvectors, which are associated with each eigenvalue, provide information about how the vectors are transformed under the transformation.
Example:
Consider the transformation matrix:
The eigenvalues of A are 1 and 4.
The eigenvector corresponding to 1 is the vector (1, 0), which indicates that vectors in the direction of the first eigenvector are not changed under the transformation.
The eigenvector corresponding to 4 is the vector (0, 1), indicating that vectors in the direction of the second eigenvector are stretched by a factor of 4.
Therefore, the eigenvalues 1 and 4 of the transformation represent stretching and contraction, respectively