eigenvectors

Summary

Characteristic polynomial

Finding eigenspaces

Eigenvalues of triangular matrices

• are the diagonal entries
• extends from determinant of triangular matrices

Invariant with transpose

Equivalent statements of invertibility

Eigenvalues of 2x2 matrices

Concept

Eigenvalues and eigen vectors

• premultiplying a matrix to a vector has the same effect as scaling it

vectors that do not change direction during transformation by the matrix, 3b1b video

Characteristic polynomial

Eigenvalue of 0

means that some values are squished to 0 space, decreasing dimension

Algebraic multiplicity

• the exponent associated with a root of the characteristic polynomial

Eigenspace and geometric multiplicity

• set of eigenvectors associated with a particular eigenvalue

Eigenvalues of 2x2 matrices

Application

Eigenvalues and eigenspaces

Fibonnacci

Extra

Octave

# Coefficients of the characteristic polynomial
poly(A)

# Eignevalues
roots(poly(A))
eig(A)