IntermediateTopic 54 of 104Engineering Mathematics
Eigenvalues and eigenvectors
Av=λv identifies directions a transformation only stretches; they give natural frequencies and stability.
In depth
If Av=λv with v≠0, λ is an eigenvalue and v an eigenvector. Find λ from det(A-λI)=0. Eigenvalues give natural frequencies of structures, growth rates, stability of systems and principal stresses.
Key points
- Trace = sum of eigenvalues
- Determinant = product
- Real symmetric matrices have orthogonal eigenvectors
ExampleThe matrix [[2,0],[0,3]] has eigenvalues 2 and 3.
Eigenvectors show directions a system only stretches, and eigenvalues give natural frequencies, growth rates and stability margins.
What is Eigenvalues and eigenvectors?
Av=λv identifies directions a transformation only stretches; they give natural frequencies and stability.
Why is it important to understand eigenvalues and eigenvectors?
Eigenvectors show directions a system only stretches, and eigenvalues give natural frequencies, growth rates and stability margins.
What does a negative eigenvalue mean in a dynamic system?
In a linear system x dot equals A x, a negative real eigenvalue means that mode decays. A positive real eigenvalue means it grows and the system is unstable.
How are eigenvalues found in practice?
Solve the characteristic equation det(A minus lambda I) equals zero for small matrices. For larger ones, use numerical algorithms in software.
How is Eigenvalues and eigenvectors used in practice?
Typical applications include computing the natural frequencies of a building from its mass and stiffness matrices, checking whether a control system is stable from the eigenvalues of its state matrix and finding principal stresses at a point in a solid.
What should I learn before Eigenvalues and eigenvectors?
Study "Vector spaces and basis" first (topic 53 of 104 in Engineering Mathematics). It covers the groundwork this topic builds on.
What are the key points of Eigenvalues and eigenvectors?
Trace = sum of eigenvalues. Determinant = product. Real symmetric matrices have orthogonal eigenvectors.
Can you give an example of Eigenvalues and eigenvectors?
The matrix [[2,0],[0,3]] has eigenvalues 2 and 3.
What comes after Eigenvalues and eigenvectors?
The next topic is "Diagonalisation", which builds directly on this one.
How difficult is Eigenvalues and eigenvectors?
It is classed as Intermediate in Engineering Mathematics (topic 54 of 104). It is an intermediate topic. Work through the beginner topics in Engineering Mathematics first.