Eigen Values Flashcards
What is a hermitian matrix A^*.
A = (aij) => A^* = (conjugate aij)^T.
So ( 1 i )
1 + i, 1 - 2i )
= ( 1 1 - i )
-i 1 + 2i )
What does skew-symmetric / antisymmetric matrix mean?
A^T = -A
State the fundamental theorem of algebra.
Every p(x) ∈ ℂ[x], p(x) != 0, of degree n has precisely n complex roots, when counted with their multiplicity.
Define eigenvalue and eigenvector.
Let A ∈ Mn(F). A scalar λ is an eigenvalue of A is ∃ v ∈ F^n, v != 0n, s.t.
Av = λv
v is an eigenvector for the eigenvalue λ.
Define spectrum of A and how it is denoted.
Set of all eigenvalues of A.
σ(A)
T or F. If A ∈ Mn(R), then the eigenvalues of A must be strictly within R.
False, they can also be complex numbers or any set that R/F is a proper subset of.
T or F. If v is a eigen-vector of A then kv is also an eigenvector of A.
True.
How do we compute eigenvalues? Let A ∈ Mn(F).
det(A - λIn) = 0 where λ is the e-value.
Roots of characteristic polynomial.
What is the characrteristic (a) polynomial (b) equation of A?
(a) Pa(x) = det(A-xIn)
(b) det(A-xIn) = 0
Define eigenspace of A. A ∈ Mn(F).
The set of λ eigenvectors of A union with the zero vector.
Eλ := N(A - λIn)
T or F. Let A ∈ Mn(F) and let λ ∈ σ(A) then Esubλ is a subspace of F^n.
True.
Distinguish between algebraic multiplicity and geometric multiplicity.
gsubx, the geometric multiplicity of λ is the dimension of Esubλ.
asubλ, the algebraic multiplicity of λ is the multiplicity of λ as a root of the characteristic polynomial Pa(x).
State the important result relating to asubλ and gsubλ.
For all A ∈ Mn(F0 and for all λ ∈ σ(A)
1 <= gsubλ <= asubλ <= n
Define similar matrices.
Give proper notation!
Let A,B ∈ Mn(F). A is similar to B if ∃ an invertible matrix C ∈ Mn(F) s.t.
A = C^-1BC
A ~ B
T or F. If A ~ B then B ~ A.
True.
T or F. If A ~ B and B ~ C then A ~ C.
True.
T or F. A !~ A.
False.
What can we say about their determinants and traces of A ~ B?
det(A) = det(B)
trace(A) = trace(B)
Define trace(A).
The sum of the diagonal elements of the matrix A.
T or F. Let A ~ B, let Pa(x) be the characteristic polynomial of A and let Qa(x) be the characteristic polynomial of A. Pa(x) !- Qa(x).
False. They are always the same.
T or F. If A ~ B, then their e-values are the same.
True.
Let A ∈ Mn(R) or Mn(C). We have σ(A), can we calculate the trace and/or the determinant?
det(A) = λ1λ2…λn
trace(A) = λ1 + λ2 + … + λn
T or F. A ∈ Mn(R) of Mn(C) is invertible if and only if 0 !∈ σ(A).
True.
We know that A is an upper triangular matrix how can we get its e-values.
Values along diagonal are e-values.
If k ∈ N, λ ∈ σ(A), v ∈ Esubλ \ {0}, what can we say about λ^k?
It is also an e-value of A^k and v ∈ Esubλ^k of A
T or F. Non square matrices can have eigenvalues.
False.
T or F. Every eigenvalue has at least one eigenvector.
True.
Let p(x) ∈ R[x] be a real polynomial of degree n, what can we say about it?
- If z ∈ C is a root of p(x), z conjugate is a root of p(x).
- If n is odd, then p(x) has at least one real root.
- p(x) can be written as a product of real polynomials whose degree is at most 2.
T or F. All non real e-values come is complex conjugate pairs.
True.
Let A be a real symmetric matrix or a complex hermitian matrix, what can we say?
The e-values of A are all real.
When are all the e-values of A imaginary?
When A is a real skew-symmetric matrix.
Trick for solving polynomials of degree >= 3.
The only integer roots of the polynomial are factors of its constant term.
T or F. If we take the e-values of A, and find the non-zero e-vectors corresponding to these eigenvalues. These e-vectors are linearly dependent.
False.
T or F. If we take the e-values of A (and they are distinct), and find the non-zero e-vectors corresponding to these eigenvalues. These e-vectors are linearly dependent.
True.
Define diagonalizable.
Let A ∈ Mn(F), there exists B ∈ Mn(F) s.t. B^-1AB = diagonal matrix in Mn(F).
T or F. Let A ∈ Mn(F), A can be diagonalizable even if the e-vectors of A do not form a basis of F^n.
False.
T or F. If A has n distinct e-values that are all in F, then A is diagonalizable.
True.
T or F. If A is diagonalizable, then A must have distinct e-values.
False.
Define eigenbasis of F^n for A.
A basis of F^n consisting of e-vectors of a matrix A ∈ Mn(F)
Let (v1,…,vn) be an e-basis of F^n for A. Let B = [v1,…,vn]. Then B is invertible and B^-1AB = what? What is the matrix B?
diagonal matrix whose diagonal entries are the e-values of A.
The CBM, from basis (e1,…,en) of F^n to the e-basis (v1,…,vn) of F^n for A.
What can you say about (B^-1AB)^m?
= B^-1A^mB
Let A,C ∈ Mn(F) with equal characteristic polynomials, and Pa(x) has n different roots, what can we say?
Let D be the diagonal matrix with entries of e-values.
there exists B s.t. B^-1AB = D
and B^-1CB = D
Then there exists E s.t.
E^-1AE = C. So A ~ C.
Let A ∈ Mn(F) with e-values λ1,…,λn all in F. Let r be a positive integer, what are e-values of A^r?
λ1^r,…,λn^r
Let A ∈ Mn(F) with distinct e-values λ1,…,λr of A assumed to be in F. Let B1, … , Br be bases of the e-spaces Esubλ1,…,Esubλr. What can we say about these bases?
If i != j
Bi ∩ Bj = ∅
and B1 U … U Br is a linearly independent subset of F^n.
What are two conditions for A to be diagonalizable? Let A ∈ Mn(F).
(a) the sum of the algebraic multiplicities of all e-values of A in F is equal to n.
(b) the geometric multiplicity of each e-value equals its algebraic multiplicity.
Summarize diagonalization algorithm.
Compute e-values.
1. They must all be in F or we stop.
- Compute e-space and a basis for each. If all e-values are distinct continue. If all e-values not distinct but the following hold continue:
(a) the sum of the algebraic multiplicities of all e-values of A in F is equal to n.
(b) the geometric multiplicity of each e-value equals its algebraic multiplicity.
Any other case stop.
- The set B = Bλ1 U … U Bλr is an e-basis of F^n for A. Let B ∈ Mn(F) be the matrix whose columns are the vectors in B such that the first gλ1 columns are the vectors in Bλ1, then
B^-1AB = diag(λ1,…λ1,λ2,…,λ2,…,λr,…,λr).
where each n 1<=n<=r has gλ1 copies, gλ2 copies gλr copies
List effect on determinant for different EROs.
Adding scalar multiple of another row = *1
Row Switch = *-1
Scalar multiplication = *scalar
Let B = kA. We have det(A), how can we get det(B)?
det(B) = k^n * det(A)
If |σ(A)| = 1 then? If |σ(A)| = n then?
For each state e-values, algebraic multiplicity and characteristic polynomial.
A has just one e-value λ and aλ = n. Pa(x) = (x - λ)^n.
A has n distinct e-values λ1,…,λn and aλ1 = … = aλn = 1. Pa(x) = (x - λ1)(x - λ2)…(x - λn)
