Eigenvalues and eigenvectors describe how a linear transformation acts on special directions. They are useful for understanding repeated transformations, geometric changes, and many applications in science and data analysis.
A linear transformation T maps vectors in a vector space to other vectors in that same space. Usually, a transformation can rotate, stretch, shrink, or shear a vector. An eigenvector is a nonzero vector whose direction is unchanged by the transformation, except that its direction may be reversed.
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What is an eigenvector?
A nonzero vector that a linear transformation maps to a scalar multiple of itself, so its direction is preserved or reversed.
What is an eigenvalue?
The scalar λ in T(v)=λv; it tells how the corresponding eigenvector is scaled.
Why is the zero vector excluded from the definition of eigenvector?
The zero vector has no direction, so it cannot represent a direction preserved by a transformation.
What does a negative eigenvalue indicate geometrically?
The vector is scaled and points in the opposite direction along the same line.
How can eigenvectors help explain a repeated process?
They identify patterns that retain their direction, while eigenvalue magnitudes indicate whether those patterns grow, shrink, or persist under repeated application.
Quiz questions
For a square matrix A, which equation's roots give its eigenvalues?Answer: det(A−λI)=0
For an eigenvalue λ, what does its geometric multiplicity measure?Answer: The dimension of the eigenspace associated with λ.
When is an n×n matrix diagonalizable?Answer: When it has n linearly independent eigenvectors.
What is true of every eigenvalue of a real symmetric matrix?Answer: It is real, although it need not be an integer.
In principal component analysis, what does an eigenvalue of a covariance matrix represent for its eigenvector direction?Answer: The variance associated with that principal direction.