ELEC4631 Chap.2 State-Space Models: Eigenstructure, Realisations and Similarity
State-Space Models: Eigenstructure, Realisations and Similarity
The opening block supplies the language used by every later chapter. A state is the smallest internal description that, together with future input, determines future motion. Continuous-time LTI dynamics are written x-dot=Ax+Bu and y=Cx+Du. Eigenvalues identify modal rates; eigenvectors identify modal directions; algebraic multiplicity comes from the characteristic polynomial and geometric multiplicity from a null space. A matrix is diagonalisable only when every eigenvalue supplies enough independent eigenvectors. Symmetric quadratic forms connect eigenvalue signs to definiteness, preparing Lyapunov theory. State-space realisations are not unique: under the lecture convention z=Tx, the matrices become TAT⁻¹, TB, CT⁻¹ and D. Similarity preserves eigenvalues and transfer function, while matching transfer functions alone does not prove two realisations are similar or internally minimal.
What this chapter covers
- 01Choosing sufficient independent states
- 02Dimensions and roles of A, B, C and D
- 03Characteristic polynomial and eigenvectors
- 04Algebraic versus geometric multiplicity
- 05Diagonalisability and defective Jordan structure
- 06Symmetric quadratic forms and definiteness
- 07Transfer-function realisations and z=Tx similarity
Repeated eigenvalue with one missing direction
- polynomialThe characteristic polynomial is (lambda-2)(lambda+1)^2. Thus 2 has algebraic multiplicity one and -1 has algebraic multiplicity two.
- first spaceFor lambda=2, ker(A-2I)=span{[1,0,0]^T}; geometric multiplicity is one.
- second spaceFor lambda=-1, A+I forces the first and third vector components to zero. The eigenspace is span{[0,1,0]^T}.
- comparisonThe repeated eigenvalue -1 has geometric multiplicity one, smaller than algebraic multiplicity two.
- basisThere are only two independent eigenvectors for a three-state matrix, so no invertible eigenvector matrix V exists.
- conclusionConclude A is not diagonalisable and identify -1 as the failed eigenvalue rather than giving an unsupported label.
Key terms
- State
- A sufficient internal coordinate vector: current state plus future input determines future evolution under the model.
- Algebraic multiplicity
- The repetition count of an eigenvalue in det(lambda I-A). It says nothing by itself about how many eigenvectors exist.
- Geometric multiplicity
- The dimension of ker(A-lambda I). It counts independent eigenvectors for that eigenvalue.
- Diagonalisable
- A has an invertible eigenvector basis V so A=VΛV⁻¹. Equivalently, geometric and algebraic multiplicities agree for every eigenvalue.
- Similarity
- An invertible coordinate change. Under z=Tx, A_z=TAT⁻¹, B_z=TB and C_z=CT⁻¹; eigenvalues and transfer function are preserved.
- Quadratic form
- A scalar xᵀQx with symmetric Q. Eigenvalue signs of Q classify positive, negative, semidefinite or indefinite curvature.
State-Space Models: Eigenstructure, Realisations and Similarity FAQ
Is every repeated eigenvalue defective?
No. Repetition is algebraic multiplicity. A repeated eigenvalue is defective only when its eigenspace has smaller dimension. A scalar multiple of I has a repeated eigenvalue and a full eigenvector basis; a Jordan block has fewer independent eigenvectors.
Why split a cross-term across Q?
For symmetric Q, the two off-diagonal contributions combine: q12 x1x2 + q21 x2x1 = 2q12 x1x2. Therefore a scalar cross-term coefficient is divided equally between symmetric entries. Testing a nonsymmetric transcription can obscure the intended definiteness.
Do equal transfer functions prove similarity?
No. They prove the same zero-state input-output map. A nonminimal realisation may contain an uncontrollable or unobservable mode cancelled from the transfer function. Similarity is stronger and requires an invertible T satisfying all transformed matrix relations.
Which coordinate convention does the guide use?
The lecture convention z=Tx. Hence x=T⁻¹z and the transformed matrices are TAT⁻¹, TB, CT⁻¹ and D. Other texts may define x=Tz; state the convention so the algebra remains internally consistent.
What does a complete eigenstructure answer look like?
Begin with c_A(lambda)=det(lambda I-A), factor it and state each root with algebraic multiplicity. For every repeated eigenvalue, write A-lambda I, row-reduce if needed and give a basis for its null space. The dimension of that basis is geometric multiplicity. Compare a_lambda and g_lambda explicitly for every root, sum the available independent eigenvectors and then state diagonalisability. If the question continues to dynamics, connect each eigenvalue to its exponential rate and each eigenvector to a modal direction; a missing vector forecasts a Jordan polynomial factor. Use trace and determinant as an independent root check, not as a replacement for the characteristic equation. This order mirrors both the current Quiz 1 subparts and the older reference-answer presentation while keeping all numbers in the new problem.
How do modelling, quadratic forms and similarity connect?
A physical model chooses coordinates that close the first-order dynamics. Quadratic forms then measure size or energy in those coordinates, and a similarity transform changes the coordinate description without changing the underlying trajectories or eigenvalues. Under z=Tx, a quadratic x^TPx becomes z^T(T^{-T}PT^{-1})z; positivity is preserved under the invertible congruence. This is why later Lyapunov certificates and LMIs can be moved between coordinates while their matrix entries change. A strong response distinguishes coordinate-dependent objects—entries of A, B, C, P and the numerical state trajectory—from invariants such as eigenvalues, transfer function under similarity, controllability, observability and definiteness. It also checks units: adding position and velocity without scaling may be algebraically allowed in a chosen state but can make an energy interpretation questionable.
How can I use an AI tutor without losing the course method?
Ask for a fresh matrix with a specified feature—two equal eigenvalues but a full eigenspace, a defective block, or a symmetric indefinite quadratic form. Solve it on paper first. Give the tutor your characteristic polynomial, null-space basis and conclusion, and ask it to identify the first inconsistent line rather than to replace the solution. For modelling, describe the physical differential equation and ask it to check whether your state is sufficient and your A/B/C/D dimensions agree. Keep generated examples outside graded work and follow the current analytical-working requirement: the final product of practice is your derivation, not a software transcript.
Exam move
Practise three blank-page routines. First, convert a second-order physical equation to first-order state space, name the states and check every matrix dimension. Second, take a matrix with a repeated eigenvalue and write the complete characteristic-polynomial/null-space/multiplicity conclusion; include the actual eigenspace basis because it will become a modal direction. Third, verify a proposed similarity transform using A_zT=TA, B_z=TB and C_zT=C before computing inverses. Add one quadratic-form drill: build the symmetric matrix from scalar cross-terms and classify it through eigenvalues or principal minors. Use trace and determinant as cheap eigenvalue checks. Do not move into matrix exponentials until you can tell, from multiplicities, whether diagonalisation is available or a Jordan contribution must appear.
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