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Linear Algebra Eigenvalues and Diagonalization cheat sheet - grade college

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Math Grade college

Linear Algebra Eigenvalues and Diagonalization Cheat Sheet

A printable reference covering eigenvalues, eigenvectors, characteristic polynomials, diagonalization, eigenspaces, and matrix powers for college.

TopicsAlgebraVectorsAlgorithmsFunctionsMathematics
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Eigenvalues and diagonalization explain how a square matrix acts along special directions that only get stretched or reversed. This cheat sheet helps students connect computations with the structure of linear transformations. It is useful for solving systems, finding matrix powers, analyzing stability, and understanding change of basis.

Key Facts

  • A scalar λ\lambda is an eigenvalue of AA if there is a nonzero vector v\mathbf{v} such that Av=λvA\mathbf{v} = \lambda \mathbf{v}.
  • The characteristic polynomial of an n×nn \times n matrix AA is p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I) or equivalently det⁡(λI−A)\det(\lambda I - A), depending on convention.
  • Eigenvalues are found by solving det⁡(A−λI)=0\det(A - \lambda I) = 0.
  • The eigenspace for an eigenvalue λ\lambda is Eλ=Null⁡(A−λI)E_{\lambda} = \operatorname{Null}(A - \lambda I).
  • An n×nn \times n matrix AA is diagonalizable if it has nn linearly independent eigenvectors.
  • If A=PDP−1A = PDP^{-1}, then the columns of PP are eigenvectors of AA and DD contains the corresponding eigenvalues on its diagonal.
  • If A=PDP−1A = PDP^{-1}, then Ak=PDkP−1A^k = PD^kP^{-1} for any positive integer kk.
  • The algebraic multiplicity of λ\lambda is its multiplicity as a root of p(λ)p(\lambda), and the geometric multiplicity is dim⁡(Eλ)\dim(E_{\lambda}).

Vocabulary

Eigenvalue
An eigenvalue λ\lambda is a scalar for which Av=λvA\mathbf{v} = \lambda \mathbf{v} has a nonzero solution v\mathbf{v}.
Eigenvector
An eigenvector is a nonzero vector v\mathbf{v} whose direction is unchanged by the transformation AA, so Av=λvA\mathbf{v} = \lambda \mathbf{v}.
Characteristic Polynomial
The characteristic polynomial is p(λ)=det⁡(A−λI)p(\lambda) = \det(A - \lambda I) or p(λ)=det⁡(λI−A)p(\lambda) = \det(\lambda I - A), and its roots are the eigenvalues.
Eigenspace
The eigenspace EλE_{\lambda} is the set of all vectors satisfying (A−λI)v=0(A - \lambda I)\mathbf{v} = \mathbf{0}.
Diagonalizable Matrix
A matrix is diagonalizable if it can be written as A=PDP−1A = PDP^{-1} for some invertible matrix PP and diagonal matrix DD.
Geometric Multiplicity
Geometric multiplicity is the number of linearly independent eigenvectors associated with an eigenvalue, equal to dim⁡(Eλ)\dim(E_{\lambda}).

Common Mistakes to Avoid

  • Using the zero vector as an eigenvector is wrong because eigenvectors must be nonzero, even though A0=λ0A\mathbf{0} = \lambda \mathbf{0} is always true.
  • Solving A−λI=0A - \lambda I = 0 instead of det⁡(A−λI)=0\det(A - \lambda I) = 0 is wrong because eigenvalues come from a determinant equation, not entry-by-entry equality.
  • Assuming repeated eigenvalues always give enough eigenvectors is wrong because algebraic multiplicity can be larger than geometric multiplicity.
  • Mixing the order of eigenvectors in PP and eigenvalues in DD is wrong because each diagonal entry DiiD_{ii} must match the eigenvector in column ii of PP.
  • Trying to diagonalize a matrix without checking independence is wrong because PP is invertible only when its eigenvector columns are linearly independent.

Practice Questions

  1. 1 Find the eigenvalues of A=[2103]A = \begin{bmatrix}2 & 1 \\ 0 & 3\end{bmatrix}.
  2. 2 For A=[400−1]A = \begin{bmatrix}4 & 0 \\ 0 & -1\end{bmatrix}, find eigenvectors for each eigenvalue and write a diagonalization A=PDP−1A = PDP^{-1}.
  3. 3 Let A=[1101]A = \begin{bmatrix}1 & 1 \\ 0 & 1\end{bmatrix}. Find its eigenvalue and decide whether AA is diagonalizable.
  4. 4 Explain why a matrix with nn distinct eigenvalues must be diagonalizable, but a matrix with repeated eigenvalues might not be.

Understanding Linear Algebra Eigenvalues and Diagonalization

The practical work begins after a possible eigenvalue has been found. Substitute that value into the matrix formed by subtracting the eigenvalue from every diagonal entry. Then row reduce the resulting system.

The free variables describe every vector in that eigenspace. A zero vector may appear as a formal solution to the system, but it never counts as an eigenvector. Students often lose valid eigenvectors by setting free variables carelessly.

Keep parameters until the end, then choose simple values to produce a basis for the eigenspace. Any nonzero multiple of an eigenvector works, so answers can look different while describing the same direction.

Repeated eigenvalues need extra attention. A root that occurs several times in the characteristic polynomial does not automatically provide the same number of independent eigenvectors. Its algebraic multiplicity counts repeated roots.

Its geometric multiplicity counts independent directions found from the null space. The geometric multiplicity can never exceed the algebraic multiplicity. For a matrix to be diagonalizable, the total number of independent eigenvectors across all eigenspaces must equal the size of the matrix.

A repeated eigenvalue is safe when its eigenspace has enough dimensions. If it does not, the matrix is called defective. This is why finding roots alone is not enough.

Diagonalization is a coordinate change. In the original coordinates, a transformation may mix the components of every vector, making repeated applications hard to track. In the eigenvector coordinate system, each coordinate changes independently by its matching eigenvalue.

The diagonal matrix records those independent scale factors. Converting into that coordinate system, applying the simple diagonal action, then converting back explains the order of the matrix product.

This idea appears in population models, where long term growth is controlled by eigenvalues, and in systems of differential equations, where positive, negative, or zero real parts indicate growth, decay, or neutral behavior. Computer graphics, data science, and vibration analysis use related coordinate changes to separate important patterns.

Matrix powers show why the size of an eigenvalue matters. Repeated multiplication raises each diagonal entry to the same power. An eigenvalue with absolute value greater than one tends to grow under repeated steps.

One with absolute value less than one tends to shrink. A negative value causes signs to alternate. For complex eigenvalues, the effect combines scaling with rotation, even when the original matrix has only real entries.

When checking work, verify each proposed eigenvector directly by multiplying it by the original matrix. Check that the number of eigenvectors matches the dimensions claimed for each eigenspace.

Keep the chosen eigenvectors in the same order as their eigenvalues when building the change of basis matrix and diagonal matrix. A mismatch in ordering is a common source of an incorrect result.