21. Eigenvalues and Eigenvectors

Introduction to Eigenvalues and Eigenvectors

This lecture introduces the concept of eigenvalues and eigenvectors, which are special numbers and vectors that are associated with square matrices. The lecture explains what eigenvectors are, how they relate to eigenvalues, and why they are important.

What is an Eigenvector?

• A matrix acts on vectors by multiplying them.

• An eigenvector is a vector that comes out in the same direction as it went in when multiplied by a matrix.

• Mathematically, this means that Ax = λx, where λ is a scalar value called the eigenvalue.

Eigenvectors with Eigenvalue Zero

• If Ax = 0x, then x is an eigenvector with eigenvalue zero.

• These eigenvectors can be found in the null space of the matrix.

Finding Eigenvectors and Eigenvalues

• We cannot use elimination to find eigenvectors and eigenvalues because we have two unknowns (λ and x) that are multiplied together.

• Determinants will be used to find these values.

Examples of Matrices with Known Eigenvectors/Eigenvalues

• Projection matrices: Only vectors that lie on the line being projected onto are eigenvectors with eigenvalue 1. All other vectors have different eigenvalues.

Eigenvectors and Eigenvalues

In this section, the speaker discusses eigenvectors and eigenvalues.

Projection Matrices

• A vector x in a plane is unchanged by P, which is a projection matrix.

• Any x in the plane is an eigenvector with an eigenvalue of one.

• The vector perpendicular to the plane is another eigenvector with an eigenvalue of zero.

Permutation Matrices

• A permutation matrix switches the two components of x.

• One eigenvector for this matrix is (1, 1) with an eigenvalue of one.

• Another eigenvector for this matrix is (-1, 1) with an eigenvalue of -1.

Special Fact About Eigenvalues

• n by n matrices will have n eigenvalues.

• An example of a special quadratic matrix that has only one non-zero eigenvalue: [0 0 0; 0 0 0; 0 0 1].

Finding Eigenvalues and Eigenvectors

In this section, the speaker explains how to find eigenvalues and eigenvectors by solving Ax = λx.

Solving Ax = λx

• To solve Ax = λx, we can bring it onto the same side and rewrite it as A - λI = 0.

• If there is a non-zero x that satisfies this equation, then A - λI must be singular.

• The determinant of A - λI must be zero for A - λI to be singular.

• After finding the eigenvalue(s), we can find the eigenvector(s) by solving for x using elimination.

Special Properties of Matrices

In this section, the speaker discusses how special properties of matrices affect their eigenvalues.

Symmetric Matrices

• Symmetric matrices have real eigenvalues.

Trace of a Matrix

In this section, the speaker explains how to use the trace of a matrix to find its other eigenvalue.

Trace of a Matrix

• For a 2x2 matrix, if one eigenvalue is known, then its trace can be used to find the other eigenvalue.

Introduction to Eigenvalues and Eigenvectors

In this section, the instructor introduces eigenvalues and eigenvectors. He explains how they are used in linear algebra and provides an example of a 2x2 matrix.

Finding Eigenvalues

• The instructor shows how to find eigenvalues by computing the determinant of A minus lambda I.

• He simplifies the equation and sets it equal to zero.

• The instructor solves for lambda using factorization or the quadratic formula.

Finding Eigenvectors

• After finding the eigenvalues, the instructor moves on to finding eigenvectors.

• He subtracts each eigenvalue from A to get a new matrix, which he then uses to find x in Ax = lambda x.

• For one of the eigenvalues, he finds that x is [1 1].

• For the other eigenvalue, he finds that x is [-1 1].

Overall, this section provides an introduction to eigenvalues and eigenvectors with a clear example of how they can be found for a 2x2 matrix.

Eigenvectors and Eigenvalues

In this section, the speaker discusses eigenvectors and eigenvalues.

Eigenvectors and Basis

• A line of eigenvectors exists, but only one vector is needed for a line.

• The natural vector to pick as the eigenvector with lambda two is minus one one.

• If elimination is done on that vector and set the free variable to be one, we get minus one and get that eigenvector.

Adding Matrices

• When adding three I to a matrix, its eigenvectors don't change, but its eigenvalues are three bigger.

• If something is done to the matrix or if something is known about the matrix, what's the conclusion for its eigenvectors and eigenvalues?

• Simple observation: when adding a constant value to a matrix, its eigenvectors remain unchanged while its eigenvalues increase by that constant value.

Matrix Addition

In this section, the speaker discusses how matrices can be added together.

Adding Two Matrices

• Suppose there are two matrices A and B. If Ax equals lambda x and B has eigenvalue alpha, then A plus B x equals lambda plus alpha x.

• Normally the eigenvalues of A plus B or A times B are not eigenvalues of A plus eigenvalues of B because they don't add linearly.

Orthogonal Matrices

In this section, the speaker discusses orthogonal matrices and their properties.

Eigenvectors and Eigenvalues of a Rotation Matrix

• The matrix that rotates every vector by 90 degrees is an orthogonal matrix called Q.

• The sum of the eigenvalues of Q is zero, and the product of the eigenvalues is one.

• Since Q rotates every vector by 90 degrees, there are no eigenvectors that come out in the same direction they went in. This leads to trouble with eigenvalues.

Complex Eigenvalues

• The equation for the eigenvalues of Q is lambda squared plus one equals zero, which yields complex conjugate pairs as eigenvalues.

• Complex numbers can enter when dealing with perfectly real matrices like Q.

Symmetric Matrices

• If a matrix is symmetric or close to symmetric, its eigenvalues will stay real. Anti-symmetric matrices have purely imaginary eigenvalues.

Introduction to Eigenvalues and Eigenvectors

In this section, the lecturer introduces eigenvalues and eigenvectors. The lecture focuses on finding the eigenvalues of a given matrix.

Finding Eigenvalues

• The eigenvalues of a triangular matrix can be read off from its diagonal.

• To find the eigenvalues, compute the determinant of A minus lambda I, where lambda is subtracted from the diagonal elements.

• For a 2x2 matrix with repeated eigenvalue three, both lambda one and lambda two are equal to three.

Finding Eigenvectors

• To find eigenvectors, we look for vectors x in the null space of A minus lambda I.

• For a 2x2 matrix with repeated eigenvalue three, there is only one independent eigenvector.

Conclusion

• Matrices with repeated eigenvalues may not have enough independent eigenvectors.