MAST10007 Chap.10 Eigenvalues, Eigenvectors and Diagonalisation
Eigenvalues, Eigenvectors and Diagonalisation
Chapter 10 develops eigenvalues, eigenvectors and diagonalisation for University of Melbourne MAST10007. An eigenvector is a nonzero direction that a linear transformation only scales. Eigenvalues solve det(A-λI)=0; each eigenspace is the null space of A-λI. Diagonalisation requires a full basis of eigenvectors, not merely a list of eigenvalues.
Over the real numbers, some matrices fail because their eigenvalues are complex; over the complex numbers the picture can change, though defective matrices still exist. It pairs exact definitions with a new completed example, two concept diagrams, diagnostic contrasts and answered retrieval prompts.
The workflow separates candidate values from valid directions.
The characteristic equation locates eigenvalues, but every eigenvector must then be found from a homogeneous system and must be nonzero. Repeated eigenvalues require an eigenspace-dimension check because multiplicity in the polynomial does not automatically provide enough independent vectors. A diagonalisation record must keep eigenvectors and diagonal entries in matching order.
Verification through the matrix relation confirms that the assembled basis and scaling data describe the original transformation.
What this chapter covers
- 01
Eigenvector meaning
- 02
Characteristic equation
- 03
Finding eigenspaces
- 04
Algebraic multiplicity
- 05
Geometric multiplicity
- 06
Diagonalisation criterion
- 07
Building P and D
- 08
Real versus complex fields
- 09
Repeated eigenvalues
- 10
Verification and failure modes
- 11
Matching eigenvector columns with diagonal entries
- 12
Checking eigenspace dimension for repeated values
- 13
Verifying diagonalisation through the defining matrix relation
Eigenvalues, Eigenvectors and Diagonalisation worked example
- stepThe triangular entries give eigenvalues 4 and 2. For λ=4 choose v1=(1,0); for λ=2 solve 2x+y=0 and choose v2=(1,-2).
- stepThe eigenvectors are independent, so P=[[1,1],[0,-2]] and D=diag(4,2).
- stepBecause AP=PD, A=PDP^{-1}. Consequently A^k=PD^kP^{-1} with D^k=diag(4^k,2^k).
- stepThe check AP=PD is quick and avoids a sign error in P; it also verifies that eigenvectors and eigenvalues occupy matching columns.
Key terms
- Eigenvector meaning
- An eigenvector is a nonzero vector whose image remains on its span: Av=λv. The scalar λ may reverse direction, shrink, stretch, or annihilate the vector. Zero is excluded because it satisfies the equation for every λ and carries no direction.
- Characteristic equation
- Eigenvalues solve det(A−λI)=0. The determinant equation is available only for square matrices. Factoring the characteristic polynomial gives candidates; each must still be paired with an eigenspace.
- Finding eigenspaces
- For a fixed λ, solve (A−λI)v=0. The solution space is the eigenspace, and every nonzero vector in it is an eigenvector. A basis for that null space is more informative than one chosen vector.
- Algebraic multiplicity
- Algebraic multiplicity counts how often λ occurs as a root of the characteristic polynomial. It is not the number of visibly repeated diagonal entries unless the matrix is triangular.
- Geometric multiplicity
- Geometric multiplicity is the dimension of the eigenspace. It is at least one and at most the algebraic multiplicity. A shortfall is precisely what can prevent diagonalisation.
- Diagonalisation criterion
- An n×n matrix is diagonalizable over the chosen field exactly when it has n linearly independent eigenvectors. Distinct eigenvalues guarantee independence, but repeated eigenvalues require an eigenspace-dimension check.
- Eigenspace
- For a fixed eigenvalue, the eigenspace is the null space of the shifted matrix. It contains the zero vector and all eigenvectors associated with that value. Solving the homogeneous system reveals the available independent eigenvector directions.
- Geometric multiplicity
- Geometric multiplicity is the dimension of an eigenspace. It cannot exceed the corresponding algebraic multiplicity. Diagonalisation succeeds only when the combined eigenspaces supply enough independent eigenvectors to form a basis of the whole space.
Eigenvalues, Eigenvectors and Diagonalisation FAQ
How do I check eigenvector meaning?
An eigenvector is a nonzero vector whose image remains on its span: Av=λv. The scalar λ may reverse direction, shrink, stretch, or annihilate the vector. Zero is excluded because it satisfies the equation for every λ and carries no direction. Its characteristic equation λ²+1=0 has no real eigenvalues.
What is the main trap in eigenvalues, eigenvectors and diagonalisation?
The zero vector is never an eigenvector. Repeated eigenvalues do not automatically provide enough eigenvectors. Keep the same column ordering in P and D.
Can AI help with this MAST10007 topic?
Yes. Ask Sia to explain one step, create a fresh practice problem, or check your reasoning. Use it to learn rather than complete graded assessment, and follow University of Melbourne academic-integrity rules.
Why do repeated eigenvalues need an additional check?
A repeated root of the characteristic polynomial records algebraic multiplicity, not the number of independent eigenvectors. You must solve the shifted homogeneous system and determine the eigenspace dimension. Too few independent directions prevents diagonalisation.
How can I verify that my matrices really diagonalise the original matrix?
Keep each eigenvector column aligned with its eigenvalue on the diagonal and check the defining multiplication relation. This verifies every column equation at once. Also confirm that the eigenvector matrix is invertible, since dependent columns cannot form the required basis.
Exam move
Use the varied 10-page chapter as a dependency map: define each object, reproduce the fresh worked example, explain both diagrams, answer the two retrieval prompts, and perform an independent check. For eigenvalues, eigenvectors and diagonalisation, revisit the first line where your representation, dimensions, or theorem conditions diverge.
Practise the pipeline in separate labelled stages: characteristic equation, eigenvalue list, shifted null spaces, independent eigenvector count, ordered construction, and verification. For repeated values, pause before claiming diagonalisation and compare algebraic with geometric multiplicity. Finish by explaining why diagonal form simplifies repeated powers instead of treating the final formula as an isolated trick.
Working through Eigenvalues, Eigenvectors and Diagonalisation in MAST10007? Sia is AskSia’s AI Mathematics tutor — ask any MAST10007 Eigenvalues, Eigenvectors and Diagonalisation question and get a clear, step-by-step explanation grounded in how MAST10007 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
