Concepts / Eigenvalues

Eigenvalues

A symmetric matrix is positive definite when every eigenvalue is positive.

  • Programming

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.

Matrix Aacts onλusame direction as uEigenvector unonzeroEigenvalue λscaling valuetransformsscaled by λdetermines scale
What happens to a nonzero eigenvector when a matrix acts on it, and how does the eigenvalue describe that change?

Testing an Eigenvector

Definition

A nonzero vector u is an eigenvector of a matrix A when the result of applying A to u can be written as one scalar multiple of u. The defining relationship is Au = λu, where λ is the eigenvalue associated with u.

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.

Worked example

Checking a Candidate Direction

Suppose a matrix sends a nonzero vector u to 4u. Decide whether u is an eigenvector and identify its eigenvalue.

  1. 1
    Compare the output with u

    The output is a scalar multiple of the original vector: Au = 4u.

  2. 2
    Match the defining relationship

    The expression has the form Au = λu.

  3. 3
    Read the scalar

    The scalar multiplying u is 4, so the associated eigenvalue is 4.

Result

u is an eigenvector, and its eigenvalue is 4.

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.

Symmetric matrixEvery eigenvaluepositivePositive definiteSymmetric matrixEvery eigenvaluenonnegativePositive semidefiniteandclassifies asandclassifies as
What changes between positive definite and positive semidefinite matrices, and where does the symmetry requirement enter the definition?
ClassificationSymmetry in the supplied definitionEigenvalue conditionAre zero eigenvalues allowed?
Positive definiteRequiredEvery eigenvalue is positiveNo
Positive semidefiniteRequiredEvery eigenvalue is nonnegativeYes

The difference is whether the eigenvalue condition excludes zero.

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.

Formula

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.

Ueigenvector columnsAsymmetric matrixDeigenvalues on diagonalUᵀtranspose of Ucombined withcombined withcombined with
How is a symmetric matrix reconstructed from its eigenvectors and eigenvalues?
Worked example

Reading the Factors

In A = UDUᵀ, identify what the factors represent for a symmetric matrix.

  1. 1
    Read U

    The columns of U are the matrix’s eigenvectors, organized as an orthonormal basis.

  2. 2
    Read D

    D is diagonal, and its diagonal entries are the eigenvalues paired with those eigenvectors.

  3. 3
    Interpret the product

    The product represents A through its eigenvector directions and the associated scaling values.

Result

The spectral decomposition is a structural description of A, not merely a product of unrelated matrices.

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.

EigenvectordirectionsNonzero eigenvaluesRange of Aspanned by pairedeigenvectorsZero eigenvaluesNull space of Aspanned by pairedeigenvectorsRank of Acount of nonzeroeigenvaluesseparate by valueseparate by valuepaired directions spancount determinespaired directions span
How do eigenvalue directions with nonzero values contribute to the range, while zero-eigenvalue directions form the null space?

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

  • Common mistake

    Treating nonnegative as equivalent to positive.

    What it looks like

    Classifying a symmetric matrix with a zero eigenvalue as positive definite.

    Why it fails

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

    Do this instead

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

  • Common mistake

    Leaving symmetry out of the supplied definition.

    What it looks like

    Checking only the signs of eigenvalues without noting that the definition is for a symmetric matrix.

    Why it fails

    The supplied definition explicitly includes symmetry.

    Do this instead

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

  • Common mistake

    Calling the zero vector an eigenvector.

    What it looks like
    Using u = 0 in Au = λu.
    Why it fails

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

    Do this instead

    An eigenvector must be nonzero.

  • Common mistake

    Assuming every eigenvector contributes to the range.

    What it looks like

    Including a zero-eigenvalue direction among the directions that span the range.

    Why it fails

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

    Do this instead

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

Check Your Understanding

MEDIUM

A symmetric matrix has eigenvalues 5, 2, and 0. Classify it as positive definite or positive semidefinite, explain whether zero is allowed in that classification, and state what the zero-eigenvalue direction contributes to the matrix structure.

Hints
  • Compare the eigenvalue signs with the definitions.
  • Positive definite excludes zero, while positive semidefinite allows it.
  • Use the relationship between zero eigenvalues and the null space.
  1. 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.

Key Takeaways

  • An eigenvector is a nonzero vector satisfying Au = λu.
  • For a symmetric matrix, every positive eigenvalue means positive definite; every nonnegative eigenvalue means positive semidefinite.
  • A symmetric matrix can be represented as A = UDUᵀ, with eigenvectors in U and eigenvalues on the diagonal of D.
  • The number of nonzero eigenvalues equals rank, and their eigenvectors span the range.
  • Zero-eigenvalue eigenvectors span the null space, while SVD is introduced only as a decomposition technique in the supplied material.

Related Topics

Related concepts

More concepts starting with E