CHAPTER 5 Eigenvectors and Eigenvalues

§ 5.1 Eigenvectors and Eigenvalues

THEOREM 1

The eigenvalues of a triangular matirx are the entries on its main diagonal.

THEOREM 2

if v₁,...,v_r are eigenvectors that correspoind to distinct eigenvalues λ₁,...,λ_r of an n⨉n matrix A, then the set {v₁,...,v_r} is linearly independent.

§ 5.2 The Characteristic Equation

determinant

det A = (-1)^r · (product of pivots in U)

THEOREM

The Invertible Matrix Theorem (continued)

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Let A be an n ⨉ n matrix. Then A is invertible if and only if:

19. The number - is not an engenvalue of A.


20. The determinant of A is not zero.

THEOREM 3

Properties of Determinants

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Let A and B be n ⨉ n matrices.

1. A is invertible if and only if det A ≠ 0.


2. det AB = (det A)(det B).


3. det A^T = det A.


4. If A is triangular, then det A is the product of the entries on the main diagonal of A.


5. A row replacement opoeration on A does not change the determinant. A row interchange changes the sign of the determinant. A row sacling also scales the determinant by the same scalar factor.

characteristic equation

det (A - λI) = 0

similar

A is similar B if there is an invertible matrix P such that P^-1AP = B or equivalently, A = PBP^-1

THEOREM 4

If n ⨉ n matrices A and B are similar, then they have the same characteristic polynomial and hence the same eigenvalues (with the same multiplicities).

§ 5.3 Diagonalization

THEOREM 5

The Diagonalization Theorem
An n ⨉ n matirx A is diagonalizable if and only if A has n linearly independent eigenvectors.
In fact, A = PDP^-1, with D a diagonal m if and only if the columns of P are n linearly independent eigenvectors of A. In this case, the diagonal entries of D are eigenvalues of A taht correspond, respectively, to the eigenvectors in P.

THEOREM 6

An n ⨉ n matrix with n distinct eigenvalues is diagonalizable.

THEOREM 7

Let A be an n ⨉ n matirx whose distinct eigenvalues are λ₁,...,λ_p.

1. For 1 ≤ k ≤ p, the dimension of the eigenspace for λ_k is less than or equal to the multiplicity of the eigenvalue λ_k.


2. The matrix A is diagonalizable if and only if the sum of the dimensions of the distinct eigenspaces equals n, and this happens if and only if the dimension of the eigenspace for each λ_k equals the multiplicity of λ_k.


3. If A is diagonalizable and B_k is a basis for the eigenspace corresponding to λ_k for each k, the the total collection of vectors in the sets B₁,...,B_p forms an eigenvector basis for Rⁿ.