Chapter 5 Eigenvectors and Eigenvalues

What are the steps to diagonalization?

1) Find the eigenvalues of A. 2) Find linearly independent eigenvectors of A (if they exist) 3) Construct the matrix P whose columns are the above eigenvectors 4) Construct the diagonal matrix D from the corresponding eigenvalues of D (in the same order as step 3) 5) Check AP=PD

What is an Eigenvector?

A non-zero vector such that when multiplied by a matrix A we get a scalar multiple of that vector. In other words, it's a vector x such that Ax=λx where λ is some scalar. Note that an eigenvector must be nonzero, by definition. (pg. 269

What is an Eigenvalue?

A scalar λ is called an eigenvalue if there is a nontrivial solution to the equation Ax=λx (pg. 269)

What does it mean for a square matrix to be diagonalizable?

A square matrix is diagonalizable if it is similar to a diagonal matrix (pg. 284)

Diagonalization theorem. State and Prove

An nxn matrix A is diagonalizable if and only if A has n linearly independent eigenvectors. In fact A=PDP⁻¹, with D a diagonal matrix, iff 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 (pg. 284.)

Theorem 6: an nxn matrix with n distinct eigenvalues is...

An nxn matrix with n distinct eigenvalues is diagonalizable

T/F: Row reduction can be used to find eigenvalues.

F: An echelon form of a matrix A usually does NOT display the eigenvalues of A (pg. 270)

T/F: A number c is an eigenvalue of A if and only if the equation Ax=cx has only the trivial​ solution, and Ax=cx and ​(A−​cI)x=0 are equivalent equations.

F: A number c is an eigenvalue of A if and only if the equation Ax=cx has NONTRIVIAL solutions, and Ax=cx and ​(A−​cI)x=0 are equivalent equations.

T/F: An nxn matrix is invertible iff it's determinant is zero.

F: An nxn matrix is invertible iff it's determinant is NOT zero (pg. 277)

T/F: If an nxn matrix has less than n distinct eigenvalues then it is not diagonalizable

F: It is still possible for an nxn matrix to have less than n eigenvalues and still be diagonalizable (see theorem 7)

T/F: If two square matrices of the same size have the same eigenvalues, then they must be similar

F: This is only true if they share the same eigenvectors

T/F: If Ax=λx for some vector x​, then λ is an eigenvalue of A.

F: x must be nonzero by definition of eigenvector

T/F: Given any linear transformation T from R^n to R^m, there exists an nxm matrix A, called the standard matrix of T. Left hand multiplication by A will convert vectors in R^n to vectors in R^m.

False: Given any linear transformation T from R^n to R^m, there exists an MxN matrix A, called the standard matrix of T. Left hand multiplication by A will convert vectors in R^n to vectors in R^m (pg. 290)

T/F: The dimension of the eigenspace for an eigenvalue is greater than or equal to the multiplicity of the eigenvalue.

False: the dimension of the eigenspace for an eigenvalue is LESS then or equal to the multiplicity of the eigenvalue (pg. 287)

What is an Eigenspace corresponding to an eigenvalue λ?

If A is an nxn matrix, the eigencspace corresponding to an eigenvalue λ, is the null space of the matrix (A-λI). In other words, it's the set of all solutions to the equation (A-λI)x=0. The eigenspace consists of the zero vector and all the eigenvectors corresponding to λ (pg. 270)

Prove that if A is diagonalizable and invertible, then so is A⁻¹

If A is​ diagonalizable, then A=PDP⁻¹ for some invertible P and diagonal D. IF A is invertible, then 0 is not an eigenvalue and the diagonal entries of D are nonzero and thus D is invertible. It follows then that A⁻¹=(PDP⁻¹)⁻¹=PD⁻¹P⁻¹ and so we see that A⁻¹ is diagonalizable (OHW 5.3.27)

If two matrices are similar? What can we say about their characteristic polynomial? (Theorem 4)

If two matrices are similar, then they have the same characteristic polynomial and thus have the same eigenvalues

What does it mean for two matrices to be similar?

Matrices A and B are similar iff there exists an invertible matrix such that P⁻¹AP=B

If B is an Eigenspace of of an nxn matrix, then B is a subspace of what?

R^n

Theorem 2: State and prove.

Recall: One way to prove the statement "If P then Q" is to show that P and the negation of Q leads to a contradiction. This strategy is used in the proof of the theorem (pg. 272).

T/F: An nxn matrix A is invertible iff zero is NOT an eigenvalue of A

T: an nxn matrix A is invertible iff zero is NOT an eigenvalue of A (pg. 277)

What is the characteristic equation of an nxn matrix? Why is it important?

The characteristic equation of an nxn matrix A is the scalar equation det(A-λI)=0. We care about the characteristic equation because it helps us find the eigenvalues of A

What is the characteristic polynomial?

The characteristic polynomial of a matrix is det(A-λI) (pg. 278)

Theorem 1: State and prove. The eigenvalues of a triangular matrix...

The eigenvalues of a triangular matrix are the entries on its main diagonal (pg. 271)

What is the space denoted by C^n?

The space of n-tuples of complex numbers. (pg. 5.5)

T/F: When an nxn matrix A is real, it's complex eigenvalues occur in conjugate pairs.

This is true. See the picture for the proof (pg. 300)

T/F: IF A is diagonalizable and Bk is the eigenspace corresponding to λk for each k, then the total collection of vectors is the sets B₁,...,Bk forms an eigenvector space of R^n

True (pg. 287)

T/F: A matrix A is not invertible if and only if 0 is an eigenvalue of A

True. If 0 is an eigenvalue of​ A, then there are nontrivial solutions to the equation Ax=0x. The equation Ax=0x is equivalent to the equation Ax=0​, and Ax=0 has nontrivial solutions if and only if A is not invertible.

T/F: The matrix A is diagonalizable iff the sum of the dimensions of the eigenspaces equals n.

True: in face, this happens iff the characteristic polynomial factors completely into linear facters and if the dimension of the eigenspace for each eigenvalue equals the multiplicity of that eigenvalue (pg. 287)

Theorem 7: Let A be nxn whose distinct eigenvalues are λ₁,...,λp...

pg. (287)

Review of some properties of determinants. (Theorem 3)

pg. 278

Explain what a complex eigenvalue and a complex eigenvector are

pg. 297

Explain complex vectors and their conjugates

pg. 299

Properties of conjugates applied to vectors and matrices

pg. 300