Key Definitions Section 9 Flashcards Preview

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Flashcards in Key Definitions Section 9 Deck (9)
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1
Q

Linear transformation is diagonalisable

A

If there is a basis B for V s.t. MatB,B(T) is a diagonal matrix

2
Q

Square matrix is diagonalisable

A

There is an invertible matrix P s.t. P-1AP is diagonal

3
Q

Algebraic multiplicity

A

Largest power k s.t. (x-lambda)^k is a factor of the character polynomial k

4
Q

Geometric multiplicity

A

Dimension g_lambda of the eigenspace

5
Q

Monic polynomial

A

Polynomial where the leading coefficient is 1

6
Q

Minimum polynomial

A

Monic polynomial Mt(x) with coefficients in F of smallest degree s.t. Mt(T) = 0

7
Q

Theorem 9.1.5 Equivalence

A

(i) T is diagonalisable
(ii) There is a basis for V consisting of eigenvectors for T
(iii) The characteristic polynomial Ct(x) is a product of linear factors and a_lambda = g_lambda for all eigenvalues
(iv) The minimum polynomial Mt(x) is a product of distinct linear factors

8
Q

Easy way to show that a matrix is diagonalisable/not diagonalisable

A
  • Find the eigenvalues of T

- Show the algebraic and geometric multiplicities are not the same

9
Q

Relationship between linear factors and diagonalisability

A

Linear factors -> does not mean diagonalisable
Non linear factors -> non diagonalisable
Diagonalisable -> linear factors