Diagonalization

In state-space control systems, the state equations are often coupled, meaning each state variable depends on others. Diagonalization is a transformation technique that decouples these equations by converting the system matrix into a diagonal form, making analysis and solution significantly simpler. This concept is directly tied to eigenvalue decomposition and is frequently tested in GATE.

Core Concept of Diagonalization

A linear time-invariant system in state-space form is given as dx/dt = Ax + Bu, y = Cx + Du. When the system matrix A is non-diagonal, the state equations are coupled, meaning the rate of change of x₁ depends on x₂, x₃, and so on. Solving such coupled systems analytically is complex. Diagonalization

The transformation uses the modal matrix P, whose columns are the linearly independent eigenvectors of A. When we substitute x = Pz into the state equation and multiply both sides by P⁻¹, we get dz/dt = P⁻¹APz + P⁻¹Bu = Λz + P⁻¹Bu. Here Λ is the diagonal matrix containing the eigenvalues of A on its main diagonal and zeros everywhere else.

This transformation is valid only when A has n linearly independent eigenvectors, which is guaranteed when all eigenvalues are distinct. For repeated eigenvalues, the Jordan canonical form is used instead of a fully diagonal form. In GATE problems, distinct eigenvalues are assumed unless stated otherwise.

Mathematical Expression

Given system matrix A of size n×n, the eigenvalues λ₁, λ₂, ..., λₙ are found by solving the characteristic equation det(A - λI) = 0. For each eigenvalue λᵢ, the corresponding eigenvector pᵢ satisfies (A - λᵢI)pᵢ = 0. The modal matrix P is assembled as P = [p₁ | p₂ | ... | pₙ], and the diagonalized matrix is obtained as Λ = P⁻¹AP. The resulting Λ has the form diag(λ₁, λ₂, ..., λₙ).

In the decoupled z-domain, each scalar equation is żᵢ = λᵢzᵢ + (i-th row of P⁻¹B)u. The solution is simply zᵢ(t) = e^(λᵢt) zᵢ(0) + integral term, which is a first-order exponential response. The state transition matrix in the original coordinates becomes Φ(t) = e^(At) = P e^(Λt) P⁻¹, where e^(Λt) = diag(e^(λ₁t), ..., e^(λₙt)).

Practical Understanding

Diagonalization has direct physical meaning. Each decoupled mode zᵢ represents an independent natural mode of the system. The eigenvalue λᵢ governs how fast or slow that mode responds. In a stable system, all eigenvalues must have negative real parts, meaning each mode decays exponentially. In an unstable system, at least one eigenvalue has a positive real part, and that mode grows unboundedly.

From a control design perspective, diagonalization simplifies controller design. Once the system is decoupled, each mode can be controlled independently. This is the basis of modal control or pole placement in state-space design. It also makes the computation of the matrix exponential straightforward, which is critical for solving the state equation analytically.

Solved Numerical Example

Consider a second-order system with matrix A = [[0, 1], [-2, -3]]. The eigenvalues are found from det(A - λI) = λ² + 3λ + 2 = 0, giving λ₁ = -1 and λ₂ = -2. The corresponding eigenvectors are found by substituting each eigenvalue back. For λ₁ = -1: (A + I)p₁ = 0 gives p₁ = [1, -1]ᵀ. For λ₂ = -2: (A + 2I)p₂ = 0 gives p₂ = [1, -2]ᵀ. The modal matrix P and diagonal matrix Λ are then formed as shown.

Given:
A = [[0, 1], [-2, -3]]  (2×2 system matrix)

Why this formula applies:
Characteristic equation gives eigenvalues; eigenvectors form modal matrix P.

Formula:
det(A - λI) = 0  →  λ² + 3λ + 2 = 0
P⁻¹AP = Λ

Substitution:
λ² + 3λ + 2 = (λ+1)(λ+2) = 0
λ₁ = -1,  λ₂ = -2

For λ₁ = -1:  (A + I)p = 0  →  p₁ = [1, -1]ᵀ
For λ₂ = -2:  (A + 2I)p = 0  →  p₂ = [1, -2]ᵀ

Calculation:
P = [[1, 1], [-1, -2]]
P⁻¹ = [[-2, -1], [1, 1]]
Λ = P⁻¹AP = [[-1, 0], [0, -2]]

Final Answer:
Λ = diag(-1, -2)  — diagonal matrix with eigenvalues on main diagonal
Both eigenvalues negative → system is stable

Exam Tip: In GATE, if a question gives A and asks for system response or stability, directly compute eigenvalues using det(A - λI) = 0. For diagonalization, remember that Λ = P⁻¹AP and the state transition matrix is e^(At) = P·diag(e^(λ₁t), e^(λ₂t))·P⁻¹. Repeated eigenvalues indicate Jordan form, not diagonal form.

• The characteristic equation det(A - λI) = 0 yields n eigenvalues, which are placed on the diagonal of Λ.
• Each eigenvector pᵢ satisfies (A - λᵢI)pᵢ = 0 and becomes a column of the modal matrix P.
• The transformation x = Pz converts the coupled state equation into dz/dt = Λz, where each mode evolves independently.
• The state transition matrix simplifies to e^(At) = P · diag(e^(λ₁t), ..., e^(λₙt)) · P⁻¹.
• System stability is directly read from eigenvalues: all eigenvalues must have negative real parts for asymptotic stability.

Quick Revision

• Diagonalization transforms coupled state equations dx/dt = Ax into decoupled form dz/dt = Λz using x = Pz.
• Eigenvalues λᵢ are found from det(A - λI) = 0; eigenvectors form columns of modal matrix P.
• Diagonal matrix: Λ = P⁻¹AP, with eigenvalues on diagonal and zeros elsewhere.
• State transition matrix: e^(At) = P · e^(Λt) · P⁻¹, where e^(Λt) = diag(e^(λ₁t), e^(λ₂t), ...).
• Valid only when eigenvectors are linearly independent (distinct eigenvalues). Repeated eigenvalues require Jordan form.
• Exam trap: Do not confuse the modal matrix P (eigenvector matrix) with the transformation matrix in Jordan form.
• Stability check: All eigenvalues with Re(λᵢ) < 0 implies asymptotic stability of the original system.