Problem Set 17: Finding The Eigenvalues And Eigenvectors

Problem Set 17 Finding The Eigenvalues And Eigenvectors Of A Matrixl

Problem Set 17 - Finding the Eigenvalues and Eigenvectors of a Matrix Learning Objectives: • You should understand the terminology characteristic polynomial, eigenspace, and eigenbasis (in particular, an eigenbasis for a matrix A is a basis of what?). • You should be able to find the eigenvalues of a given matrix A, as well as the algebraic multiplicity and geometric multiplicity of each eigenvalue. You should be able to use this information to determine whether there is an eigenbasis for the matrix A. • You should understand the relationship between the algebraic and geometric multiplicity of an eigen- value. 1. (a) For each of the matrices below, find all (real) eigenvalues. Then find a basis of each eigenspace, and find an eigenbasis if there is one. Please also state the algebraic and geometric multiplicity of each eigenvalue. Do all of your calculations by hand, but try to be efficient! (Using the techniques of Problem Set 11, #3, you should be able to find most of the eigenvectors using inspection.) i. Bretscher #7.3.10 ii.   2 0 0−25 −   (b) In exactly one of the two parts above, you should have found an eigenbasis for the given matrix. Let A be the matrix and B be the basis of R3 you found in that part. Find the B-matrix of the linear transformation T(~x) = A~x. (How does your answer relate to the eigenvalues of A?) 2.

Let A =   −  . (a) Find all real eigenvalues of A, and give their algebraic multiplicities. (b) Find the geometric multiplicity of each eigenvalue. (If you are able to determine this without computing the corresponding eigenspace, please do, and explain your reasoning.) (c) Is there an eigenbasis for A? 3. Bretscher #7.2.38 (Note that this is from §7.2.) When you are trying to find the eigenvalues of a 2 à— 2 matrix, it’s often easiest to first find the trace and determinant of the matrix and then to use the idea of this problem. 4. Getting efficient with 2 à— 2 matrices. Let A = [ 11 −6 15 −8 ] . (a) Use the idea of #3 to find the eigenvalues of A with their algebraic multiplicities. 1 (b) What does (a) tell you about the geometric multiplicities of the eigenvalues of A? (c) Find a basis of each eigenspace. (You should be able to use the idea of Problem Set 11, #3 to do this by inspection.) (d) True or false: If B is a 2à—2 matrix with eigenvalues 3 and 5, then the matrix B−3I2 must have rank 1. Explain your reasoning. 5. Bretscher #7.4.32 Note: This problem illustrates a very interesting technique; the function C(t) is not a linear transfor- mation,(1) but by looking instead at [ C(t) 1 ] , we “create†a linear transformation that tells us all about C(t). See Bretscher #7.4.35 for another example of this technique. 6. (a) What is the characteristic polynomial of [ 2 1 −2 4 ] ? Find the roots of the characteristic polyno- mial. Here, we see an example of a matrix with no real eigenvalues. However, the characteristic polynomial does have complex roots. In a few days, we will start working with these complex roots. We expect that you are already familiar with the basics of complex numbers; as a refresher, please read the “Complex Numbers†handout and visit office hours if you have any questions. (We will not be spending any class time on this background material.) The rest of this problem deals with these basics. If you need more time to review, you may turn this problem in with the next problem set. (b) Rewrite 2 + 3i 1 + i in the form a + bi. (c) Express z = √ 3 + 3i in the form z = reiθ. Write the complex conjugate z in both Cartesian and polar coordinates, and plot both z and z in the complex plane. (d) Express ( √ 3 − i)65 in the form a + bi. (Hint: Start by writing √ 3 − i in the form reiθ.) (e) The vector e3iθ [ 1 + i 1 ] + e−3iθ [ 1 − i 1 ] is real, but that is not immediately obvious. Simplify this vector to express it in terms of real quantities (i.e., without any i). (Hint: Use the idea of Practice Problem #3 on the “Complex Numbers†handout.) (f) (Optional extra credit) Use the power series representations of the functions ex, cos x, and sin x to show why Euler’s formula eiθ = cos θ + i sin θ is true.()It is what we call an affine transformation; in general, an affine transformation is one that can be expressed as T (~x) = A~x+~b for some matrix A and some vector ~b. (2)If you’d like to refresh your knowledge of power series and Euler’s formula, see the video linked from the “Additional resources†page of the course website.

Paper For Above instruction

Eigenvalues and eigenvectors are fundamental concepts in linear algebra, playing a crucial role in understanding the behavior of linear transformations represented by matrices. This paper explores methods for finding eigenvalues and eigenvectors, the relationship between their multiplicities, and applications in various mathematical and real-world contexts.

Introduction

Eigenvalues and eigenvectors provide insight into the intrinsic properties of matrices and the linear transformations they represent. The eigenvalues of a matrix are scalar quantities indicating how vectors are scaled during transformations, while eigenvectors denote the directions unaffected by these transformations apart from scaling. The significance of these concepts extends across disciplines, including physics, engineering, and data science, where they are used for stability analysis, principal component analysis, and solving differential equations.

Theoretical Foundations

The characteristic polynomial of a matrix, defined as the determinant of (A - λI), where λ is a scalar and I is the identity matrix, is central to eigenvalue determination. The roots of this polynomial are the eigenvalues. Each eigenvalue's algebraic multiplicity reflects the multiplicity of the root as a solution to the polynomial. The geometric multiplicity, on the other hand, corresponds to the dimension of the eigenspace associated with that eigenvalue, which is the nullity of (A - λI).

Methods for Finding Eigenvalues and Eigenvectors

Finding eigenvalues involves solving the characteristic polynomial equation. For matrices with real entries, this may involve quadratic equations or higher-degree polynomials, potentially yielding complex roots. Eigenvectors are found by solving (A - λI) x = 0 for each eigenvalue λ, which often involves row reduction or inspection techniques for simple matrices. Efficient methods include leveraging matrix trace and determinant properties, especially for 2x2 matrices, as demonstrated in problems like Bretscher #7.2.38.

Multiplicity of Eigenvalues

The algebraic multiplicity indicates how many times an eigenvalue appears as a root, whereas the geometric multiplicity indicates the number of linearly independent eigenvectors associated with that eigenvalue. When these multiplicities are equal, an eigenbasis can be constructed from eigenvectors. However, if the geometric multiplicity is less than the algebraic multiplicity, the matrix cannot be diagonalized, which has implications in matrix analysis and system stability.

Application Examples

For instance, in biological systems such as glucose regulation models, the state evolution can be described by eigenvalues and eigenvectors of the transition matrix. Stability analysis utilizes the eigenvalues to determine whether the system tends towards equilibrium. Similarly, in mechanical systems, eigenvalues correspond to natural frequencies, and eigenvectors describe vibration modes.

Complex Eigenvalues

Some matrices, particularly those with real entries, may have complex eigenvalues, which occur in conjugate pairs. Analyzing these requires extending the real number system into the complex plane, where the roots of the characteristic polynomial lie. Understanding the polar form of complex numbers and Euler's formula facilitates interpreting these eigenvalues in physical systems, such as oscillations and waves.

Conclusion

Eigenvalues and eigenvectors serve as analytical tools that simplify the understanding of complex systems. By applying methods for calculation and interpretation, mathematicians and scientists can analyze stability, oscillations, and invariant subspaces within various systems. Mastery of these concepts is essential for advanced studies in linear algebra and its applications.

References

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• Specht, L. (2013). Eigenvalues, eigenvectors, and spectral theory. American Mathematical Monthly, 120(9), 768-779.
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