For an ODE, eigensolutions of some expression \(x’=Ax\) consists of the class of solutions which remains in a line through the origin, which consists of the family which:

\begin{equation} x(t) = ke^{\lambda t} v \end{equation}

where \(\lambda\) is an eigenvalue of \(A\), and \(v\) its corresponding eigenvector.

motivation

\begin{equation} y’ = F(y) \end{equation}

an autonomous ODE, suppose we have some solution \(y=a\) for which \(y’ = 0\), that is, \(F(a) = 0\), we know that the system will be trapped there.

The eigenspace of \(T, \lambda\) is the set of all eigenvectors of \(T\) corresponding to \(\lambda\), plus the \(0\) vector.

constituents

• \(T \in \mathcal{L}(V)\)
• \(\lambda \in \mathbb{F}\), an eigenvalue of \(T\)

requirements

\begin{equation} E(\lambda, T) = \text{null}\ (T - \lambda I) \end{equation}

i.e. all vectors such that \((T- \lambda I) v = 0\).

where, \(E\) is an eigenspace of \(T\).

additional information

sum of eigenspaces is a direct sum

\(E(\lambda_{1}, T) + … + E(\lambda_{m}, T)\) is a direct sum.

see Extended Kalman Filter

The Elastic Modulus is a measurement of how much deformation takes place given some force on the system. Formally, it is the slope of the stress-strain curve, defined by:

\begin{equation} E = \frac{stress}{strain} \end{equation}

The units in pascals as it is: force per area (pascals) divided by deformation (dimensionless, as it is a fraction of old shape over new shape \(\frac{V}{V}=1\)).

Depending on how its measured, it is called different things:

• Young’s Modulus: tensile elasticity—tendency for object to deform along an axis with force applied (usually that is just called the Elastic Modulus)
• Shear’s Modulus: shear elasticity—tendency of an object to shear (deform in shape with the constant volume) with force applied
• Bulk Modulus: volumetric elasticity—tendency for an object to deform in all directions when uniformly loaded

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