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Flashcards in Chapter 6 Deck (7)
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1
Q

Diagonalisable

Definition

A

-if for a square matrix A there exists an invertible matrix P such that P^-1AP is diagonal then we say that A is diagonalisable

2
Q

How to find P for a square matrix A such that

P^-1AP = D

A
  • find the eigenvalues and eigenvectors of the matrix A
  • D is a square matrix with all zero entries except the main diagonal whose values are the eigenvalues of A i.e. d(11) = λ1, d(22) = λ2, d(33) = λ3 etc.
  • P is a matrix whose columns are composed of the eigenvectors of A, P = (v1 v2 v3 …)
3
Q

How can you tell that a matrix is not diagonalisable?

A

-if there are not enough linearly independent eigenvectors to form a square matrix, P, then we cannot diagonalise the matrix A

4
Q

Eigenspace

A

-let A be an nxn square matrix with eigenvalue λ
-we call
Uλ = {v| Av=λv, v∈ℝ^n}
the eigenspace of λ
i.e. Uλ is the set of eigenvectors with eigenvalue λ for A together with the 0 vector

5
Q

Eigenspaces and Subspaces

A

-the eigenspace Uλ is a subspace of ℝ^n

6
Q

Eigenvectors of A and Linearly Independence

A
  • suppose A is an nxn square matrix with at least m distinct eigenvalues with corresponding eigenvectors
  • then the set {v1, v2, …, vm} is linearly independent
  • proof by induction
7
Q

Diagonalisable Matrices, Calculating A^m

A
P^-1*A*P = D
A = P*D*P^-1
A^m = (P*D*P^-1)
A^m = P*D^m*P^-1