Repeated Eigenvalues, Fundamental Matrices, and Nonhomogeneous Systems

Two gaps remain in the constant-coefficient theory. A repeated eigenvalue can fail to supply enough eigenvectors, leaving the eigenpair method short of solutions; and a forcing term $g (t)$ makes the system nonhomogeneous. The connecting idea for both is the fundamental matrix: package a fundamental set as the columns of one matrix, and the whole solution theory compresses into matrix algebra. In the constant-coefficient case that matrix turns out to be a matrix exponential.

Fundamental matrices

Arrange a fundamental set of solutions side by side as the columns of a matrix.

The general solution collapses to one matrix-vector product. Writing $x = c_{1} x^{(1)} + \dots + c_{n} x^{(n)}$ as $Ψ (t)$ acting on the constant vector $c = (c_{1}, \dots, c_{n})^{⊤}$ gives

x = Ψ (t) c .

For an initial value problem $x (t_{0}) = x^{0}$ , the constant vector must satisfy $Ψ (t_{0}) c = x^{0}$ , so $c = Ψ^{- 1} (t_{0}) x^{0}$ and

x (t) = Ψ (t) Ψ^{- 1} (t_{0}) x^{0} .

A fundamental matrix (written M here, the Psi of the text) stacks independent solution vectors as columns; acting on a constant vector c it gives the general solution, and on an initial state it propagates it forward.

Among all fundamental matrices, one is distinguished by a convenient initial value. Reserve the symbol $Φ (t)$ for the fundamental matrix with

Φ (t_{0}) = I .

Its columns are the solutions with initial vectors the standard basis vectors $e^{(1)}, \dots, e^{(n)}$ . With $Φ$ the IVP solution is as simple as it can be, since $Φ^{- 1} (t_{0}) = I$ :

x (t) = Φ (t) x^{0} .

$Φ (t)$ is a transformation that carries the initial state $x^{0}$ into the state $x (t)$ at any later time — a propagator. Once $Φ$ is known, every initial condition is solved by a single matrix multiplication.

The matrix exponential

The scalar problem $x^{'} = a x$ , $x (0) = x_{0}$ , has solution $x = x_{0} e^{a t}$ . The system $x^{'} = A x$ , $x (0) = x^{0}$ , has solution $x = Φ (t) x^{0}$ . The parallel is exact once $Φ (t)$ is identified as an exponential of the matrix $A$ .

Define $exp (A t)$ by the same power series that defines the scalar exponential, with $A$ in place of $a$ and $I$ in place of $1$ :

exp (A t) = I + A t + \frac{A ^{2} t ^{2}}{2 !} + \dots + \frac{A ^{n} t ^{n}}{n !} + \dots = I + n = 1 \sum \infty \frac{A ^{n} t ^{n}}{n !} .

Every entry of this matrix series converges for all $t$ . Differentiating term by term factors out one $A$ :

\frac{d}{d t} exp (A t) = A exp (A t), exp (A \cdot 0) = I .

So $exp (A t)$ solves the same matrix initial value problem $Φ^{'} = A Φ$ , $Φ (0) = I$ that defines the special fundamental matrix. By uniqueness they are the same object:

The exponential shares the algebra of the scalar exponential, including the propagator law $exp (A t) exp (A s) = exp (A (t + s))$ and the inverse $exp (A t)^{- 1} = exp (- A t)$ . When $A$ has a full set of eigenvectors the series is not the practical route: diagonalizing $A = T D T^{- 1}$ gives $exp (A t) = T exp (D t) T^{- 1}$ with $exp (D t) = diag (e^{r_{1} t}, \dots, e^{r_{n} t})$ , so the eigenpairs already compute it. The series is needed only when eigenvectors run short.

Repeated eigenvalues

An eigenvalue $ρ$ of algebraic multiplicity $m$ may have geometric multiplicity $q < m$ : fewer than $m$ independent eigenvectors, and so fewer than $m$ solutions of the pure form $ξ e^{ρt}$ . The missing solutions must be found in a new form, and the pattern copies the repeated-root case for a single second-order equation, where a double root $r$ gave the second solution $t e^{r t}$ .

Worked example. For $A = (11 - 1 3)$ the characteristic equation is $(1 - r) (3 - r) + 1 = r^{2} - 4 r + 4 = (r - 2)^{2} = 0$ ,¹ so $r = 2$ is a double eigenvalue. Solving $(A - 2 I) ξ = 0$ gives the single condition $ξ_{1} + ξ_{2} = 0$ , so there is only one independent eigenvector, $ξ = (1, - 1)^{⊤}$ . One solution is

x^{(1)} (t) = (1 - 1) e^{2 t} .

A naive second guess $x = ξ t e^{2 t}$ fails: substituting forces $ξ = 0$ . Allowing an extra constant vector alongside the $t e^{2 t}$ term,

x = ξ t e^{2 t} + η e^{2 t},

substituting into $x^{'} = A x$ and matching the coefficients of $t e^{2 t}$ and $e^{2 t}$ separately gives two conditions:

(A - 2 I) ξ = 0, (A - 2 I) η = ξ .

The first says $ξ$ is the eigenvector. The second determines $η$ : although $A - 2 I$ is singular, the right-hand side $ξ$ lies in its column space, so the equation is solvable. Here $- η_{1} - η_{2} = 1$ , and a convenient choice is $η = (0, - 1)^{⊤}$ , giving the second solution

x^{(2)} (t) = (1 - 1) t e^{2 t} + (0 - 1) e^{2 t} .

Their Wronskian is $- e^{4 t} \neq = 0$ , so ${x^{(1)}, x^{(2)}}$ is a fundamental set and the general solution is $x = c_{1} x^{(1)} + c_{2} x^{(2)}$ .

A defective double eigenvalue r chains the eigenvector v to a generalized eigenvector w (the xi and eta of the text); the second solution carries the extra t e^{rt} term.

The phase portrait for a defective double eigenvalue is an improper node. Every term carries the factor $e^{ρt}$ , and every trajectory approaches the origin tangent to the single eigendirection (for $ρ < 0$ ) or departs along it (for $ρ > 0$ ), but no trajectory settles onto a straight line — the $t e^{ρt}$ term keeps bending it.

Improper node from a defective double eigenvalue: every trajectory is tangent to the one eigendirection at the origin, bent by the t e^{rho t} term.

This defective example also shows the matrix exponential explicitly. Assembling $x^{(1)}, x^{(2)}$ as columns gives a fundamental matrix, and normalizing to $Φ (0) = I$ by $Φ (t) = Ψ (t) Ψ^{- 1} (0)$ yields the propagator

exp (A t) = Φ (t) = e^{2 t} (1 - t t - t 1 + t),

the polynomial-in- $t$ entries being exactly what the eigenvector shortage produces — the series $I + A t + \frac{1}{2} A^{2} t^{2} + \dots$ does not collapse to a single exponential per direction, as it would for a diagonalizable matrix.

For higher multiplicities the chain lengthens: a triple defective eigenvalue adds a $t^{2} e^{ρt} /2$ term with a second generalized eigenvector solving $(A - ρ I) ζ = η$ . Packaged as a similarity transformation, the eigenvector and its generalized companions turn $A$ into its Jordan form $J = T^{- 1} A T$ : the eigenvalue on the diagonal, a $1$ just above it in the column of each missing eigenvector, and zeros elsewhere. The Jordan form is the reason the $t e^{ρt}$ pattern appears — $exp (J t)$ generates exactly those powers of $t$ .

Nonhomogeneous systems

Restore a forcing term and consider

x^{'} = P (t) x + g (t) .

The structure copies the single-equation case exactly: the general solution is the general solution of the homogeneous system plus any one particular solution.

Three methods find the particular solution $v$ , in increasing order of generality.

Diagonalization. When $A$ is constant and diagonalizable, $x = T y$ turns the system into $n$ uncoupled scalar equations $y_{j}^{'} = r_{j} y_{j} + h_{j} (t)$ with $h = T^{- 1} g$ , each solved by the first-order integrating factor.
Undetermined coefficients. When $A$ is constant and $g$ is polynomial, exponential, or sinusoidal, guess a matching form. The one twist versus the scalar case: if $g$ contains $u e^{λ t}$ with $λ$ an eigenvalue, the trial must include both $a t e^{λ t}$ and $b e^{λ t}$ , not $a t e^{λ t}$ alone.
Variation of parameters. The general method, needing only a fundamental matrix — no restriction on $P (t)$ or $g (t)$ .

Variation of parameters

Start from the homogeneous general solution $Ψ (t) c$ and promote the constant vector to an unknown function, $x = Ψ (t) u (t)$ . Differentiating and using $Ψ^{'} = P Ψ$ ,

Ψ^{'} u + Ψ u^{'} = P Ψ u + g ⟹ Ψ (t) u^{'} (t) = g (t) .

The homogeneous part cancels, leaving a purely algebraic condition on $u^{'}$ . Since $Ψ$ is invertible, $u^{'} (t) = Ψ^{- 1} (t) g (t)$ , and integrating gives the general solution.

Worked example. Solve the same system $x^{'} = (- 2 1 1 - 2) x + (2 e^{- t} 3 t)$ by variation of parameters.² The eigenvectors $(1, - 1)^{⊤}$ and $(1, 1)^{⊤}$ (for $- 3$ and $- 1$ ) give a fundamental matrix and its inverse

Ψ (t) = (e^{- 3 t} - e^{- 3 t} e^{- t} e^{- t}), Ψ^{- 1} (t) = \frac{1}{2} (e^{3 t} e^{t} - e^{3 t} e^{t}) .

Then $u^{'} (t) = Ψ^{- 1} (t) g (t)$ has components

u_{1}^{'} = e^{2 t} - \frac{3}{2} t e^{3 t}, u_{2}^{'} = 1 + \frac{3}{2} t e^{t} .

Integrating (the additive constants become the homogeneous part),

u_{1} = \frac{1}{2} e^{2 t} - \frac{1}{2} t e^{3 t} + \frac{1}{6} e^{3 t}, u_{2} = t + \frac{3}{2} t e^{t} - \frac{3}{2} e^{t} .

Assembling $x = Ψ u$ and absorbing a homogeneous $e^{- t}$ multiple into $c_{2}$ gives the general solution

x = c_{1} (1 - 1) e^{- 3 t} + c_{2} (11) e^{- t} + (11) t e^{- t} + (0 - 1) e^{- t} + (12) t - \frac{1}{3} (45),

the first two terms the homogeneous solution and the rest a particular solution, matching the undetermined-coefficients result above.

Variation of parameters with M the fundamental matrix (Psi in the text): substituting x = M u collapses the equation to u' = M^{-1} g, then integration and one matrix product rebuild x.

Summary

The fundamental matrix organizes the constant-coefficient theory.

Fundamental matrix. Columns are a fundamental set; $x = Ψ (t) c$ is the general solution, and the special $Φ (t)$ with $Φ (0) = I$ propagates initial states via $x = Φ (t) x^{0}$ .
Matrix exponential. For constant $A$ , $Φ (t) = exp (A t)$ , defined by the same series as the scalar exponential and computed from eigenpairs when they are complete.
Defective eigenvalues. Missing eigenvectors are replaced by generalized eigenvectors solving $(A - ρ I) η = ξ$ ; the extra solution carries $t e^{ρt}$ and the portrait is an improper node.
Nonhomogeneous systems. General $=$ homogeneous $+$ particular; variation of parameters, $u^{'} = Ψ^{- 1} g$ , finds the particular solution with only a fundamental matrix in hand.

Boyce, §7.8, Examples 1–2: the defective double eigenvalue of $(11 - 1 3)$ and the generalized-eigenvector second solution. ↩
Boyce, §7.9, Examples 1–3: the system $x^{'} = (- 2 1 1 - 2) x + (2 e^{- t} 3 t)$ solved by diagonalization, undetermined coefficients, and variation of parameters, all yielding the same general solution. ↩ ↩²

Fundamental matrices

The matrix exponential

Repeated eigenvalues

Nonhomogeneous systems

Variation of parameters

Summary

Footnotes