Eigenvectors are vectors that, when multiplied by a linear transformation, undergo a specific transformation. In other words, they are vectors that retain t...
Eigenvectors are vectors that, when multiplied by a linear transformation, undergo a specific transformation. In other words, they are vectors that retain their "shape" or magnitude under the linear transformation, while their "direction" changes.
Examples:
This means that when A transforms a vector in the direction of , it leaves it unchanged.
Similarly, the eigenvector corresponding to the eigenvalue 2 is:
Eigenvectors have important geometric and algebraic properties, including:
They form a subspace called the eigspace, which is the set of all vectors that are preserved by the linear transformation.
The dimension of the eigspace corresponds to the number of linearly independent eigenvectors.
The eigenvalue corresponds to the rate of stretching or compression of vectors in the eigspace.
The eigenvectors and eigenvalues form a set of orthogonal vectors that forms a base for the vector space