Matrix Measure (Logarithmic Norm)

• Matrix Measure (Logarithmic Norm) is a functional on square matrices that encapsulates the rate of solution growth or decay in linear systems.
• It offers explicit formulas for common vector norms such as 1-, 2-, and infinity-norms, facilitating error control and stability assessments.
• The concept extends to stochastic systems via the stochastic logarithmic norm, providing computable criteria for mean and mean-square stability.

The matrix measure, also known as the logarithmic norm, is a functional on square matrices associated with a chosen vector norm. First introduced by G. Dahlquist in 1958, the matrix measure provides a tool for quantifying the rate of growth or decay of solutions to linear ordinary differential equations (ODEs) and forms the basis for error growth analysis in discretization methods. The concept extends naturally to stochastic systems, yielding the stochastic logarithmic norm for stability analysis of linear Itô stochastic differential equations (SDEs) with multiplicative noise. These objects provide computable criteria for asymptotic stability in both deterministic and stochastic linear systems, underpin numerical analysis methods, and facilitate mean- and mean-square stability assessments (0809.0062).

1. Dahlquist’s Definition and Motivation

The matrix measure $\mu(A)$ associated with a vector norm $\|\cdot\|$ in $\mathbb{R}^n$ (or $\mathbb{C}^n$) is defined by

$$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$

where $\|\cdot\|$ denotes the induced matrix norm. This definition enables direct control over the growth of the norm of solutions to the linear ODE $\dot{x} = Ax$ with initial condition $x(0) = x_0$. In particular, using the Dini derivative and the explicit form of the matrix measure, the norm of the solution satisfies

$$\frac{d^+}{dt} \|x(t)\| \leq \mu(A) \|x(t)\|,\quad \Rightarrow\quad \|x(t)\| \leq e^{\mu(A)t} \|x(0)\|$$

Thus, a negative logarithmic norm, $\mu(A) < 0$, guarantees asymptotic stability of the zero solution. In numerical analysis, the matrix measure provides upper bounds for the error growth in one-step methods and underpins the distinction between forward and backward error control.

2. Explicit Formulas for Common Induced Norms

Let $\|\cdot\|$0 and $\|\cdot\|$1 denote the logarithmic norm induced by the vector $\|\cdot\|$2-norm $\|\cdot\|$3. Closed-form expressions are available for standard norms:

• For $\|\cdot\|$4

$\|\cdot\|$5

• For $\|\cdot\|$6

$\|\cdot\|$7

where $\|\cdot\|$8 denotes the largest real eigenvalue of the symmetric part.

• For $\|\cdot\|$9

$\mathbb{R}^n$0

The derivations rely on first-order expansions of the matrix norm, with attention to subtleties for $\mathbb{R}^n$1, as the spectral norm involves the eigenvalues of generally non-Hermitian matrices. For $\mathbb{R}^n$2 and $\mathbb{R}^n$3, the structure of the column and row sums directly yields the result, while for $\mathbb{R}^n$4, Weyl's inequalities apply to the symmetric part of the matrix.

3. Analytical Properties

Several key properties characterize the matrix measure. For any $\mathbb{R}^n$5 and $\mathbb{R}^n$6:

• Homogeneity: $\mathbb{R}^n$7
• Subadditivity: $\mathbb{R}^n$8
• Exponential bounding: $\mathbb{R}^n$9 for $\mathbb{C}^n$0

The exponential bound is derived by considering the solution operator $\mathbb{C}^n$1 and applying iterative expansion:

$\mathbb{C}^n$2

which leads to the differential inequality $\mathbb{C}^n$3 and thus the exponential growth constraint.

4. The Stochastic Logarithmic Norm

For the linear Itô SDE

$\mathbb{C}^n$4

with constant $\mathbb{C}^n$5, Wiener process $\mathbb{C}^n$6, and initial condition $\mathbb{C}^n$7, the stochastic logarithmic norm in the $\mathbb{C}^n$8-th mean (with vector $\mathbb{C}^n$9-norm) generalizes Dahlquist’s concept. Let $$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$0. The $$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$1-th mean stochastic logarithmic norm is given by

$$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$2

The correction term $$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$3 derives from the strong order-1 Itô–Taylor expansion and ensures accuracy for SDEs. Under mild boundedness assumptions, this limit exists and provides a means to assess the mean and mean-square stability of the equilibrium.

A fundamental stability estimate for solutions holds:

$$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$4

and consequently,

$$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$5

Thus, if $$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$6, the zero solution of the SDE is asymptotically stable in the $$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$7-th mean (0809.0062).

5. Illustrative Examples and Applications

Scalar SDE

For the scalar linear SDE $$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$8, with $$\mu(A) := \lim_{h \to 0^+} \frac{\|I + hA\| - 1}{h}$$9, $\|\cdot\|$0:

$\|\cdot\|$1

Hence, mean-square stability ($\|\cdot\|$2, $\|\cdot\|$3) is characterized by $\|\cdot\|$4, in accord with classical mean-square results.

Linearized Inverted Pendulum with Noise

Consider the system

$\|\cdot\|$5

with

$\|\cdot\|$6

For $\|\cdot\|$7, $\|\cdot\|$8, it follows that

$\|\cdot\|$9

The stability condition $\dot{x} = Ax$0 leads to the requirement $\dot{x} = Ax$1, notably revealing that noise of sufficient amplitude can stabilize the upright position in mean square [(0809.0062), §11.1].

Deterministic Logarithmic Norm Example

For $\dot{x} = Ax$2,

$\dot{x} = Ax$3

This results in the growth bound $\dot{x} = Ax$4.

6. Significance in Numerical Analysis and Stability

The matrix measure is central to the analysis of solution growth in linear ODEs and the control of discretization errors in numerical integration, particularly for one-step methods. Its extension to the stochastic logarithmic norm enables analogous stability analysis in stochastic systems, yielding precise, computable stability criteria for both first and higher moments. These criteria are instrumental in the design of balanced stochastic integrators and informed assessments of system behavior under multiplicative noise.

7. Concluding Perspectives

The matrix measure and its stochastic counterpart unify the analysis of stability and quantitative solution bounds for linear ODEs and SDEs. Their closed-form expressions for standard norms, well-characterized properties, and applicability to systematic stability theorems render them indispensable in both deterministic and stochastic settings. These tools facilitate rigorous control over solution behavior, support numerical algorithm development, and elucidate phenomena such as noise-induced stabilization (0809.0062).