05 Eigenvalues and eigenvectors Flashcards Preview

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Flashcards in 05 Eigenvalues and eigenvectors Deck (11)
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1
Q

eigenvector and eigenvalue

A

An eigenvector of an n x n matrix A is a nonzero vector x such that Ax = λx for som scalar λ. A scalar λ is called an eigenvalue of A if there is a nontrivial solution x of Ax = λx; such an x is called an eigenvector corresponding to λ.

2
Q

The eigenspace of A corresponding to λ consists of the zero vector and _____.

A

The eigenspace of A corresponding to λ consists of the zero vector and all the eigenvectors corresponding to λ.

3
Q

λ is an eigenvalue of an n x n matrix A if and only if the equation

_____

has a nontrivial solution.

A

λ is an eigenvalue of an n x n matrix A if and only if the equation

(A - λI)x = 0

has a nontrivial solution.

4
Q

The eigenvalues of a triangular matrix are the entries on ____.

A

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

5
Q

A scalar λ is an eigenvalue of an n x n matrix A if and only if λ satisfies the characteristic equation

_______

A

A scalar λ is an eigenvalue of an n x n matrix A if and only if λ satisfies the characteristic equation

det(A - λI) = 0

6
Q

Similarity

A

If A and B are n x n matrices, then A is imilar to B if there is an invertible matrix P such that P-1AP = B.

7
Q

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

A

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

8
Q

A square matrix A is said to be _______ if A is similar to a diagonal matrix, that is, if A = PDP-1 for some invertible matrix P and som diagonal matrix D.

A

A square matrix A is said to be diagonalizable if A is similar to a diagonal matrix, that is, if A = PDP-1​ for some invertible matrix P and som diagonal matrix D.

9
Q

An n x n matrix A is diagonalizable if and only if A has _____ eigenvectors.

A

An n x n matrix A is diagonalizable if and only if A has n linearly independent eigenvectors.

In fact, A = PDP-1 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 that correspond, respectively, to the eigenvectors in P.

10
Q

A is in other words diagonalizable if and only if there are enough eigenvectors to form ______.

A

A is in other words diagonalizable if and only if there are enough eigenvectors to form a basis of R^n. We call such a basis an eigenvector basis of R^n.

11
Q

Steps to diagonalize matrices

A
  1. Find the eigenvalues
  2. Find n linearly indendent eigenvectors.
  3. Construct P from the eigenvectors.
  4. Construct D from the corresponding eigenvalues.