Numbers and algebra
1829
Eigenvalues
Augustin-Louis Cauchy (the word “eigen” came later, from David Hilbert)
Why does a spinning book flip over when you toss it one way, but not the others?
A pencil can balance on its point for a moment, but a hanging pendulum always settles. The difference is in a few special numbers: directions that grow or shrink without turning. Cauchy proved key results about them in 1829. Engineers now use them to find how bridges, buildings and aircraft wings vibrate.
First, make a guess
Rotate the plane by 90°. How many directions stay pointing the same way?
- Eigenvalues λ
- 1.93 and 0.37
- Meaning
- Stretches along the gold and pink lines
A matrix moves every arrow in the plane. Most arrows get turned. The special ones, along the dashed lines, only stretch or shrink: those are eigenvectors, and the stretch factor is the eigenvalue. Rotation has no such arrows at all, so its eigenvalues are complex numbers, the language of spinning.
How they figured it out
Euler, studying spinning bodies in the 1750s, found that every rigid object has three special axes, its principal axes. These are eigenvectors. In 1829 Cauchy proved that for symmetric matrices like these, the special values are always real numbers.
The word itself came later from David Hilbert, using the German “eigen”, meaning “own”: the matrix’s own values. The same mathematics now finds the natural frequencies of buildings, the stable states of machines and the main patterns in data.
The equation
Some special directions are only stretched by a matrix, never turned; λ says by how much.
See it in real life
Throw your phone (gently, onto a bed)
Toss a book or phone spinning about its longest axis: smooth. About its shortest: smooth. About the middle axis: it flips over every time. Cosmonaut Vladimir Dzhanibekov saw a wing nut do this in space in 1985. In earthquakes, buildings whose natural frequency matches the shaking suffer most.
Where engineers use it
Vibration modes
Natural frequencies of bridges, buildings and engine parts are eigenvalues of their stiffness and mass matrices.
Stability
A system is stable if all its eigenvalues have negative real parts: the same test as poles.
Principal stresses
The directions of greatest stress in a part are eigenvectors of the stress tensor.
Worked example: A diagonal matrix
- Eigenvalues 2 and 3, along the x and y axes.
Where it came from
- Shapes become equations 1637
Built on
Eigenvectors are directions in Descartes’ coordinate plane that a matrix leaves pointing the same way.
What it led to
- Balance points in space 1772
Same technique
Whether a probe at a Lagrange point drifts or holds is decided by eigenvalues.
Where the comparison stops: Lagrange found the points in 1772; the stability analysis in eigenvalue language came later.