Let V be a vector space over F and T: V β V is a linear map. What is an eigenvector?
A non-zero vector π― is an eigenvector if T(π―) = Ζπ―
Let V be a vector space over F and T: V β V is a linear map. What is the Ζ-eigenspace?
V = {π― β² V: T(π―) = Ζπ―}
= {π― β² V: (T - ΖI)π― = π}
= ker (T - ΖI)
Let V be a vector space over F and T: V β V is a linear map. How do you know the Ζ-eigenspace is a subspace of V?
Since the eigenspace is a kernel
Why do we take π― β² β^n and Ζ β² β, even if the matrix A has real components?
A real matrix might have complex eigenvectors
as real polynomials can have complex roots
What is the characteristic polynomial of an n x n matrix A?
π§_A (t) = det (A - tI)
How are the eigenvalues of A and the roots of characteristic polynomial related?
The eigenvalues of A are the roots of the characteristic polynomial
What are the eigenvalues of an upper triangular matrix?
Its diagonal entries
What is true of the determinant and characteristic polynomial for similar matrices?
They are the same
If A and B are similar, so that A = Pβ»ΒΉBP, what is π§A(t) equal to?
π§A(t) = π§B(t)
Suppose A is an n x n matrix and Ζ β² β is an eigenvalue of A. What is the geometric multiplicity of Ζ?
The dimension of the Ζ-eigenspace
Suppose A is an n x n matrix and Ζ β² β is an eigenvalue of A. What is the algebraic multiplicity of Ζ?
The multiplicity of the root Ζ in the characteristic polynomial
How are the geometric and algebraic multiplicities of Ζ related?
The geometric multiplicity of Ζ is less than or equal to the algebraic multiplicity of Ζ
If π―β, …, π―n are eigenvectors of A corresponding to distinct eigenvalues Ζβ, …, Ζn, then what does this mean for {π―β, …, π―n}?
{π―β, …, π―n} is linearly independent
When is an n x n matrix A called diagonalisable?
If it is similar to a diagonal matrix
That is, there exists a P, such that Pβ»ΒΉAP is diagonal
Let A be an n x n matrix and let V = β^n. What statements about A are equivalent?
- A is diagonalisable
- A has n linearly independent eigenvectors
- V = V_Ζβ β … β V_Ζk, where Ζβ, …, Ζk are the distinct eigenvalues of A
- For all eigenvalues Ζ, the geometric and algebraic multiplicities are equal
What is a practical way to check if a matrix is diagonalisable or not?
See if A has n linearly independent eigenvectors
A is an n x n matrix, and its characteristic polynomial has n distinct roots in β. Is A diagonalisable?
Yes
If A is an n x n matrix over β, and
f(t) = aβ + aβt + … a_m t^m is a polynomial with complex coefficients. How is f(A) defined?
f(A) = aβ + aβA + … + a_m A^m
which is an n x n matrix
What is the Cayley-Hamilton theorem?
Every square matrix A satisfies its own characteristic equation, π§A(A) = 0
What is a Jordan block matrix? (try to draw)
A square matrix of the form
J_n(Ζ) = (Ζ 1 0 … 0)
( … … 1)
(0 … … 0 Ζ)
with Ζ on the diagonal and 1 just above the diagonal (for some fixed scalar Ζ). The subscript n gives the size.
When is a square matrix A said to be in Jordan Normal Form (JNF)?
If it consists of Jordan block matrices on the diagonal, with zeros elsewhere
Is a Jordan block matrix in JNF?
Yes
What is any n x n matrix similar to?
A matrix in Jordan normal form
What is the determinant of a matrix equal to in terms of its eigenvalues?
The product of its eigenvalues
