Eigenvalues

A Matrix’s Special Directions

A matrix can change a vector’s length and direction. Eigenvectors identify the exceptional directions whose direction is preserved when the matrix acts on them. The eigenvalue records the scaling associated with that direction.

Testing an Eigenvector

The non-zero condition matters because the zero vector would satisfy A0 = λ0 for every scalar λ. It would therefore provide no meaningful information about a special direction or a particular scaling value. The definition instead uses nonzero vectors so that the preserved direction and its scaling can be identified.

Positive Definiteness by Sign

In the supplied definition, symmetry is part of the condition. A symmetric matrix is positive definite when every eigenvalue is positive. A symmetric matrix is positive semidefinite when every eigenvalue is nonnegative.

Spectral Structure of a Symmetric Matrix

For a symmetric matrix A of rank k, the spectral decomposition theorem organizes its eigenvectors into an orthonormal basis of the surrounding vector space. If the basis vectors are u1 through ud, they provide mutually perpendicular unit directions.

A = UDUᵀ

The columns of U are the eigenvectors u1 through ud. The matrix D is diagonal: its diagonal entry in position i is the eigenvalue λi associated with ui, and its off-diagonal entries are zero. The factorization reconstructs A from its eigenvector directions and their corresponding scaling values.

Range, Null Space, and Rank

The eigenvalues divide the eigenvector directions into two structural groups. The number of nonzero eigenvalues equals the rank of the matrix. Eigenvectors paired with nonzero eigenvalues span the range of A, while eigenvectors paired with zero eigenvalues span the null space of A.

This gives the decomposition a geometric and structural interpretation. Directions with nonzero eigenvalues contribute to the outputs collected in the range. Directions with zero eigenvalues are sent to the zero vector and therefore belong to the null space.

Related Decomposition Topic

The supplied material introduces Singular Value Decomposition, abbreviated SVD, as a matrix decomposition technique. At this point, the source identifies SVD as a topic alongside the review of positive definite and positive semidefinite matrices, but it does not provide the decomposition formula, name its component matrices, or describe its computational steps.

Common Classification Mistakes

• Treating nonnegative as equivalent to positive.

  Positive definiteness requires every eigenvalue to be positive, so zero is excluded.

  Fix: Use positive semidefinite when every eigenvalue is nonnegative and zero values may occur.

• Leaving symmetry out of the supplied definition.

  The supplied definition explicitly includes symmetry.

  Fix: State both parts: the matrix is symmetric, and its eigenvalues satisfy the required sign condition.

• Calling the zero vector an eigenvector.

  The zero vector satisfies the relationship for every scalar and therefore cannot identify a special direction or eigenvalue.

  Fix: An eigenvector must be nonzero.

• Assuming every eigenvector contributes to the range.

  The source associates nonzero-eigenvalue eigenvectors with the range and zero-eigenvalue eigenvectors with the null space.

  Fix: Separate eigenvector directions according to whether their eigenvalues are nonzero or zero.

Check Your Understanding

1. Eigenvalues describe the scaling of nonzero directions that a matrix preserves. For a symmetric matrix, all-positive eigenvalues give positive definiteness, while all-nonnegative eigenvalues give positive semidefiniteness. In the spectral decomposition A = UDUᵀ, U contains the orthonormal eigenvectors and D contains their eigenvalues on the diagonal. Nonzero eigenvalues determine the rank and their eigenvectors span the range; zero eigenvalues identify directions in the null space. The supplied material introduces SVD as a decomposition technique but does not specify its formula or components.