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What the lecture covers
An eigenvector is a nonzero vector that a linear transformation leaves on its own span: it may be stretched, squished, or flipped, but not rotated off its line. The corresponding eigenvalue is the scalar describing that change. In a 3D rotation, for example, an eigenvector points along the rotation axis and has eigenvalue 1. Eigenvectors offer a way to understand a transformation beyond simply reading its matrix columns.
To find eigenvalues, rewrite Av = λv as (A − λI)v = 0. A nonzero solution exists only when A − λI has determinant zero; solving this condition gives possible eigenvalues. Substituting each value back into the equation identifies its eigenvectors. Some transformations, such as a 90-degree rotation in the plane, have no real eigenvectors, while others may have too few eigenvectors to span the space. If a full basis can be chosen from eigenvectors, changing to that eigenbasis makes the transformation matrix diagonal, with eigenvalues on its diagonal. This makes repeated applications, such as raising the matrix to a high power, much easier to calculate.
Key ideas
Eigenvectors are the special directions that a transformation keeps on the same line, while their eigenvalues give the associated scaling factors.
An eigenvector of a 3D rotation identifies its axis, and its eigenvalue is 1 because the rotation preserves length.
The defining relation is Av = λv, stating that transforming an eigenvector has the same effect as scaling it by its eigenvalue.
Eigenvalues can be found by requiring det(A − λI) = 0, which allows the altered matrix to send a nonzero vector to zero.
For the opening example, the determinant condition gives eigenvalues 2 and 3, and solving for each value reveals its eigenvector directions.
Some transformations lack real eigenvectors, as shown by a 90-degree plane rotation, while a shear has eigenvectors only along its fixed axis.
When basis vectors are eigenvectors, the transformation matrix is diagonal and its diagonal entries are their eigenvalues.
If eigenvectors span the whole space, changing to an eigenbasis diagonalizes the transformation and simplifies repeated matrix operations.
Sample questions
A nonzero vector v is an eigenvector of a matrix A. Which statement must be true about the relationship between v and Av?
AAv must have the same length as v.
BAv must point in a direction that is not on the line spanned by v.
CAv lies on the line spanned by v, possibly at the origin.
DAv must be perpendicular to v.
Show answer
Correct answer: C. An eigenvector is transformed into a scalar multiple of itself, so its image remains on its span; the scalar may also map it to the origin.
A linear transformation has an eigenvector v with eigenvalue −1/2. What does the transformation do to v?
AIt points v in the opposite direction and reduces its length to half.
BIt points v in the opposite direction and doubles its length.
CIt preserves v’s length but points it in the opposite direction.
DIt preserves v’s direction and reduces its length to half.
Show answer
Correct answer: A. The negative sign reverses the vector’s direction, while the magnitude 1/2 scales its length by one half.