Linear Algebra II

Question 1

Define the determinant of a matrix

Answer 1

For a 1 x 1 matrix A = a_11, det(A) = a_11
For an nxn matrix, if A_ij is the matrix A with the ith row and jth column removed, det(A) = the sum of (-1)^(k-1).a_kkdet(A_kk)

Question 2

Define a diagonalisable matrix

Answer 2

A matrix is diagonalisatble if the map it defines by acting on vectors in R^n has a basis consisting of its eigenvectors.

Question 3

Define an eigenvalue

Answer 3

λ is an eigenvalue of T if Tv = λv for some nonzero v.

Question 4

Define an eigenvector

Answer 4

v is an eigenvector of T if v ≠ 0 and Tv = λv for some λ ∈ R

Question 5

Define the characteristic polynomial

Answer 5

The characteristic polynomial of A is det(xI - A).

Question 6

Define an eigenspace

Answer 6

The λ-eigenspace is the set of all eigenvectors associated to λ and the zero-vector.

Question 7

Define algebraic multiplicity

Answer 7

The algebraic multiplicity of an eigenvalue λ is the number of factors of x - λ in the characteristic polynomial

Question 8

Define geometric multiplicity

Answer 8

The geometric multiplicity of λ is the dimension of the λ-eigenspace