Linear algebra MOC

Eigenvectors, eigenvalues, and eigenspaces

The eigenvectors of a Linear map are the vectors with images parallel to themselves under the transformation, i.e. the vectors which remain on their span. linalg Geometrically, such vectors are never rotated by the transformation, but are scaled by some factor 𝜆, called the eigenvalue. linalg

Formally, the eigenvectors of a transformation 𝐴 are the vectors satisfying

𝐴⃗𝐯=𝜆⃗𝐯

for some scalar 𝜆. The set of all such vectors for a given 𝜆 is called an eigenspace, as it necessarily forms a vector subspace.

𝔼𝜆⁡(𝐴)={⃗𝐯∈𝑉∣(𝐴−𝜆𝐼)⃗𝐯=⃗𝟎}

The computation of eigenvalues uses the determinant

𝐴⃗𝐯=𝜆⃗𝐯⟹(𝐴−𝜆𝐼)⃗𝐯=⃗𝟎⟹det(𝐴−𝜆𝐼)=0

to form the characteristic polynomial. Once eigenvalues are known, eigenspaces can be found using Gaußian elimination on the augmented matrix [𝐴−𝜆𝐼∣⃗𝟎].

An important process is Diagonalization, by which any diagonalisable matrix is turned into a diagonal matrix with its eigenvalues as entries along the diagonal.

Properties¹

det(𝐴) =∏𝑛𝑖=1𝜆𝑖 ^P1 2. Tr(𝐴) =∑𝑛𝑖=1𝜆𝑖 ^P2 3. 𝜆𝑖(𝐴−1) =𝜆−1𝑖(𝐴) 4. 𝜆𝑖(𝐴𝖳) =𝜆𝑖(𝐴) 5. 𝜆𝑖(𝐴 +𝑘𝐼) =𝜆𝑖(𝐴) +𝑘 6. 𝜆𝑖(𝐴𝑘) =𝜆𝑖(𝐴)𝑘 7. Cayley-Hamilton Theorem — A matrix 𝐴 satisfies its own characteristic equation (by replacing 𝜆𝑛 with 𝐴𝑛). 8. Any ℓ vectors from ℓ different eigenspaces form a linearly independent set.

For 2 ×2 matrices, properties ^P1 and ^P2 allow for 3b1b’s eigenvalue trick.

Multiplicity

For each of a transformations eigenvalues we define algebraic and geometric multiplicity as follows

• Algebraic multiplicity is how many roots of the characteristic polynomial correspond to that eigenvalue. linalg
• Geometric multiplicity is the dimensions of the eigenspace dim⁡(𝐸𝜆(𝐴)) linalg

Geometric multiplicity is at most the algebraic multiplicities.¹ It is clearly impossible for an eigenspace to have dimension 0, so we have the inequality

1≤geometric multiplicity≤algebraic multiplicity

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Footnotes

1. 2022. MATH1012: Mathematical theory and methods, pp. 103–102 ↩ ↩²