7) Determinants, eigenvalues and eigenvectors Flashcards
What is
The (n − 1) × (n − 1) matrix obtained from M by deleting the i-th row and j-th column
What is the determinant of a matrix
What is Laplace Expansion Theorem
What is the minor and cofactor of a matrix
What is an upper triangle matrix
A square matrix M is called upper triangular if Mij = 0 for all i > j,
that is, if all entries below the main diagonal are 0
What is the determinant of an upper triangle matrix
What is the determinant of an identity matrix
Describe how the elementray row operations relate to the determinant
When is a matrix invertiable
If det M ≠ 0 M is invertiable
Describe the proof that a matrix is invertiable if det M ≠ 0
What is the adjugate of a sqaure matrix A
The transpose of the cofactor matrix of A
How are matrices and their adjugate related
If A is an invertiable matrix and B is the adujgate of A
If A is a matrix and B is the adjugate of A, what is AB
AB = (det A)In = BA
What is det(AB)
det(A)det(B)
Describe the proof that det(AB) = det(A)det(B)
(Don’t need to know)
If we use the LU factorization of a permutation P A of A, then what is det(P)det(A)
- det P det A = det(PA) = det(LU) = det L det U
- Where P = the row interchanges, L = the lower triangular with ones on the diagonal, and U = an upper triangular matrix
What is an eigenvector
The vector v ∈ V is an eigenvector of T with eigenvalue
λ ∈ K if v is non-zero and T(v) = λv
What is an Eigenspace
E*λ * = {v ∈ V |T(v) = λv}.
How are Eigenspaces written
What is the geometric multiplicity of an eigenvalue
The dimension of the corresponding λ-eigenspace
What is the Linear Independence of Eigenvectors
- Let T : V → V be a linear map. Suppose that v1, . . . , vk are eigenvectors of T with distinct eigenvalues λ1, . . . , λk, respectively
- Then, the set {v1, . . . vk} is linearly independent
When is a field algebraically closed
Are the set of real numbers algebraically closed
No since the polynomial x^2+1 has no real roots
Are the set of complex numbers algebraically closed
Yes
What is the algebraic multiplicity of λ (an eigenvalue
of A)
Then the power k to which the factor x − λ appears in the characteristic polynomial pA(x)
What is the trace of a matrix
If K is an algebraically closed field and λ1, . . . , λn be all the eigenvalues of A ∈ Mn(K), what is the trace and determinant of A
Describe the proof that
Describe the proof that
