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SectionPEEProperties of Eigenvalues and Eigenvectors

The previous section introduced eigenvalues and eigenvectors, and concentrated on their existence and determination. This section will be more about theorems, and the various properties eigenvalues and eigenvectors enjoy. Like a good 4×100 meter relay, we will lead-off with one of our better theorems and save the very best for the anchor leg.

SubsectionBPEBasic Properties of Eigenvalues

Proof

There is a simple connection between the eigenvalues of a matrix and whether or not the matrix is nonsingular.

Proof

With an equivalence about singular matrices we can update our list of equivalences about nonsingular matrices.

Proof

Certain changes to a matrix change its eigenvalues in a predictable way.

Proof

Unfortunately, there are not parallel theorems about the sum or product of arbitrary matrices. But we can prove a similar result for powers of a matrix.

Proof

While we cannot prove that the sum of two arbitrary matrices behaves in any reasonable way with regard to eigenvalues, we can work with the sum of dissimilar powers of the same matrix. We have already seen two connections between eigenvalues and polynomials, in the proof of Theorem EMHE and the characteristic polynomial (Definition CP). Our next theorem strengthens this connection.

Proof

Inverses and transposes also behave predictably with regard to their eigenvalues.

Proof

The proofs of the theorems above have a similar style to them. They all begin by grabbing an eigenvalue-eigenvector pair and adjusting it in some way to reach the desired conclusion. You should add this to your toolkit as a general approach to proving theorems about eigenvalues.

So far we have been able to reserve the characteristic polynomial for strictly computational purposes. However, sometimes a theorem about eigenvalues can be proved easily by employing the characteristic polynomial (rather than using an eigenvalue-eigenvector pair). The next theorem is an example of this.

Proof

If a matrix has only real entries, then the computation of the characteristic polynomial (Definition CP) will result in a polynomial with coefficients that are real numbers. Complex numbers could result as roots of this polynomial, but they are roots of quadratic factors with real coefficients, and as such, come in conjugate pairs. The next theorem proves this, and a bit more, without mentioning the characteristic polynomial.

Proof

This phenomenon is amply illustrated in Example CEMS6, where the four complex eigenvalues come in two pairs, and the two basis vectors of the eigenspaces are complex conjugates of each other. Theorem ERMCP can be a time-saver for computing eigenvalues and eigenvectors of real matrices with complex eigenvalues, since the conjugate eigenvalue and eigenspace can be inferred from the theorem rather than computed.

SubsectionMEMultiplicities of Eigenvalues

A polynomial of degree \(n\) will have exactly \(n\) roots. From this fact about polynomial equations we can say more about the algebraic multiplicities of eigenvalues.

Proof
Proof
Proof
Proof

SubsectionEHMEigenvalues of Hermitian Matrices

Recall that a matrix is Hermitian (or self-adjoint) if \(A=\adjoint{A}\) (Definition HM). In the case where \(A\) is a matrix whose entries are all real numbers, being Hermitian is identical to being symmetric (Definition SYM). Keep this in mind as you read the next two theorems. Their hypotheses could be changed to “suppose \(A\) is a real symmetric matrix.”

Proof

Notice the key step of this proof is the ability to pitch a Hermitian matrix from one side of the inner product to the other.

Look back and compare Example ESMS4 and Example CEMS6. In Example CEMS6 the matrix has only real entries, yet the characteristic polynomial has roots that are complex numbers, and so the matrix has complex eigenvalues. However, in Example ESMS4, the matrix has only real entries, but is also symmetric, and hence Hermitian. So by Theorem HMRE, we were guaranteed eigenvalues that are real numbers.

In many physical problems, a matrix of interest will be real and symmetric, or Hermitian. Then if the eigenvalues are to represent physical quantities of interest, Theorem HMRE guarantees that these values will not be complex numbers.

The eigenvectors of a Hermitian matrix also enjoy a pleasing property that we will exploit later.

Proof

Notice again how the key step in this proof is the fundamental property of a Hermitian matrix (Theorem HMIP) — the ability to swap \(A\) across the two arguments of the inner product. Notice too, that we can always apply the Gram-Schmidt procedure (Theorem GSP) to any basis of any eigenspace, along with scaling the resulting the orthogonal vectors by their norm to arrive at an orthonormal basis of the eigenspace. For a Hermitian matrix, pairs of eigenvectors from different eigenspaces are also orthogonal. If we dumped all these basis vectors into one big set it would be an orthonormal set. We will build on these results and continue to see some more interesting properties in Section OD.

SubsectionReading Questions

1

How can you identify a nonsingular matrix just by looking at its eigenvalues?

2

How many different eigenvalues may a square matrix of size \(n\) have?

3

What is amazing about the eigenvalues of a Hermitian matrix and why is it amazing?

SubsectionExercises

T10

Suppose that \(A\) is a square matrix. Prove that the constant term of the characteristic polynomial of \(A\) is equal to the determinant of \(A\text{.}\)

Solution
T20

Suppose that \(A\) is a square matrix. Prove that a single vector may not be an eigenvector of \(A\) for two different eigenvalues.

Solution
T21

Suppose that \(A\) is a square matrix of size \(n\) with an eigenvalue \(\lambda\text{.}\) For \(\lambda\in\complexes\text{,}\) prove that \(\lambda+\alpha\) is an eigenvalue of the matrix \(A+\alpha I_n\text{.}\)

  1. Construct a proof that begins by employing a vector \(\vect{x}\) that is an eigenvector of \(A\) for \(\lambda\text{.}\)
  2. Construct a proof that establishes that \(\lambda+\alpha\) is a root of the characteristic polynomial of \(A+\alpha I_n\text{.}\)
T22

Suppose that \(U\) is a unitary matrix with eigenvalue \(\lambda\text{.}\) Prove that \(\lambda\) has modulus 1, i.e. \(\modulus{\lambda}=1\text{.}\) This says that all of the eigenvalues of a unitary matrix lie on the unit circle of the complex plane.

T30

Theorem DCP tells us that the characteristic polynomial of a square matrix of size \(n\) has degree \(n\text{.}\) By suitably augmenting the proof of Theorem DCP prove that the coefficient of \(x^n\) in the characteristic polynomial is \((-1)^n\text{.}\)

T50

Theorem EIM says that if \(\lambda\) is an eigenvalue of the nonsingular matrix \(A\text{,}\) then \(\frac{1}{\lambda}\) is an eigenvalue of \(\inverse{A}\text{.}\) Write an alternate proof of this theorem using the characteristic polynomial and without making reference to an eigenvector of \(A\) for \(\lambda\text{.}\)

Solution