Eigen

What is an eigenvector?

A nonzero vector v such that Av = λv

How do you find eigenvalues?

Set up the characteristic equation: det(A - λI) = 0, then solve for λ

How do you find eigenvectors once you have eigenvalues?

Plug each λ into A - λI, set equal to 0 and solve using row reduction

What does it mean to diagonalize a matrix?

Writing A as A = PDP^-1

When is a matrix diagonalizable?

When it has enough linearly independent eignvectors to form a basis

How do you construct P and D?

D is the eigenvalues along the diagonal

P is the corresponding eigenvectors as columns

What does it mean for vectors to be orthogonal?

Their dot product is 0

u * v = 0

They are perpendicular

What is an orthogonal set?

A set of vectors where every pair is orthogonal

What is an orthonormal set?

A set of vectors that are orthogonal and each has a unit length of 1

What is the goal of Gram-Schmidt?

To convert a linearly independent set into an orthogonal (or orthonormal set)

What is the formula in Gram-Schmidt?

proj_u (v) = (v u)/(u u) u

2 vectors/matrices are considered orthogonal if the inner product or dot product is equal to?

0

Is it orthogonal if the dot product is NOT equal to 0?

No, has to be equal to 0

How do you find the unit length?

Take the square root of the dot product of each entry