Eigenvectors and eigenvalues are often found unintuitive by students.
The topic requires a solid understanding of prior concepts: matrices as linear transformations, determinants, linear systems of equations, and change of basis.
Proper visualization is crucial for understanding.
Linear Transformations in Two Dimensions
Example: A linear transformation moves basis vectors i-hat and j-hat to new coordinates, represented by a matrix.
i-hat to (3, 0)
j-hat to (1, 2)
Matrix representation: columns are [3, 0] and [1, 2]
Vectors usually get knocked off their span during transformation.
Special vectors stay on their span and are stretched or squished.
i-hat is a special vector stretched by a factor of 3.
Vector (-1, 1) is stretched by a factor of 2.
Eigenvectors and Eigenvalues
Special vectors that remain on their span are called eigenvectors.
The factor by which they are stretched or squished is the eigenvalue.
Example: Rotation in 3D
Eigenvector remains on its span, indicating the axis of rotation with eigenvalue 1.
Eigenvectors and eigenvalues give insight into the transformation beyond matrix representation.
Symbolic Representation
Expression: A * v = 位 * v, where
A is the matrix
v is the eigenvector
位 is the eigenvalue
Rewriting the equation:
(A - 位I) * v = 0, where I is the identity matrix
Determinant of (A - 位I) must be zero for non-zero solutions.*
Computational Example
Example Matrix: columns [2, 1] and [2, 3]
Finding eigenvalues involves setting determinant to zero.
Subtract 位 from diagonal elements and compute determinant.
Example: 位 = 1 makes determinant zero.
Eigenvectors: solve (A - 位I) * v = 0
Example: Matrix [3, 0] and [1, 2]
Eigenvalues: 位 = 2 and 位 = 3
Eigenvectors: e.g., (-1, 1) for 位 = 2, stretched by factor of 2.*
Special Cases
Transformations without eigenvectors (e.g., 90-degree rotation): No real eigenvalues.
Shear transformation: Has eigenvalue 1, only vectors on x-axis are eigenvectors.
Diagonal matrices: Eigenvalues on diagonal, easier for computations.
Eigenbasis
Basis vectors as eigenvectors form an eigenbasis.
Diagonal matrices in an eigenbasis simplify power computations.
Change of basis: Use eigenvectors to form new basis coordinates.
Sandwich transformation matrix with change of basis matrix.
Resulting matrix is diagonal in the new coordinate system.
Conclusion
Eigenbasis makes matrix operations more manageable.
Not all transformations have enough eigenvectors for an eigenbasis.