Finding Eigenvalues and Eigenvectors

Professor Dave Explains
17 Jul 201917:09
EducationalLearning
32 Likes 10 Comments

TLDRThe video explains the mathematical concepts of eigenvalues and eigenvectors. It defines them, shows examples of finding eigenvalues and associated eigenvectors for sample matrices, and discusses how although the eigenvector solutions represent a vector form, any scalar multiple also qualifies. The process involves finding the characteristic polynomial, solving for eigenvalues, then plugging them back in to solve for eigenvectors using row operations on the matrix.

Takeaways
  • 😀 Eigenvalues and eigenvectors have important applications in linear algebra, physics, and more
  • 😲 Eigenvectors are vectors that satisfy Ax = λx, where A is a matrix and λ is a scalar eigenvalue
  • 📉 Finding eigenvalues involves solving the characteristic equation det(A - λI) = 0
  • 📊 The solutions λ to the characteristic equation are the eigenvalues
  • ➕ Each eigenvalue has associated eigenvectors that can be found by solving (A - λI)x = 0
  • 🔢 Eigenvectors are only defined up to scalar multiples - cx is also an eigenvector if x is
  • 🎯 The number of eigenvalues for an n x n matrix A is less than or equal to n
  • 🤔 Eigenvalues represent special scalars λ linked to eigenvectors x satisfying Ax = λx
  • 🧮 By convention eigenvectors are chosen to have 1 as a component value during calculations
  • 💡 Understanding eigenvalues and eigenvectors unlocks deeper linear algebra concepts
Q & A

  • What is the definition of an eigenvector?

    -An eigenvector is a vector that, when multiplied by a square matrix A, results in that same vector multiplied by a scalar value called an eigenvalue.

  • What is the significance of eigenvalues and eigenvectors in linear algebra?

    -Eigenvalues and eigenvectors represent an important concept in linear algebra that has many applications in math, physics, and other fields. They can be used to solve systems of equations, analyze vibrations, and more.

  • How do you find the eigenvalues of a matrix?

    -To find the eigenvalues of a matrix A, you subtract the scalar lambda from the diagonal entries to get the matrix (A - λI). Then take the determinant and set it equal to 0 to get the characteristic equation. The solutions to this equation are the eigenvalues.

  • Once you have found the eigenvalues, how do you find the associated eigenvectors?

    -For each eigenvalue λ, plug it back into (A - λI) to get a new matrix. Then convert to a system of equations and use row operations to solve for the components of the eigenvector x.

  • Why can any scalar multiple of an eigenvector also be an eigenvector?

    -If x is an eigenvector, meaning Ax = λx, then for any scalar c, we have A(cx) = c(Ax) = c(λx) = λ(cx). So cx must also be an eigenvector.

  • What is the characteristic polynomial of a matrix?

    -The characteristic polynomial is the polynomial formed by taking the determinant of (A - λI). Setting this polynomial equal to 0 gives the characteristic equation, whose solutions are the eigenvalues.

  • What are some applications of eigenvalues and eigenvectors?

    -Applications include solving systems of linear differential equations, describing natural frequencies of vibrations, distinguishing quantum energy states in physics, and separating modes of motion.

  • Why can't the matrix (A - λI) be invertible when solving for eigenvectors?

    -If (A - λI) were invertible, we could multiply both sides of the eigenvector equation by its inverse and show x must equal 0. But eigenvectors must be nontrivial (not equal to the zero vector).

  • What does it mean for two eigenvectors to share the same eigenvalue?

    -Eigenvectors with the same eigenvalue represent parallel or related modes of behavior under the transformation defined by the matrix.

  • How are eigenvalues and eigenvectors useful in quantum mechanics?

    -In quantum mechanics, the different eigenvalues of an operator represent different quantized energy states, and the eigenvectors are the wavefunctions for the particles occupying those states.

Outlines
00:00
🤓 Defining Eigenvalues and Eigenvectors

Introduces eigenvalues and eigenvectors as useful linear algebra concepts with applications in math, physics, and solving systems of differential equations. Defines eigenvectors as vectors having a special relationship with matrix A where Ax = λx for some scalar λ. λ's are called eigenvalues. Shows example of verifying (1,1) is an eigenvector of a 2x2 matrix with eigenvalue -2.

05:02
😎 Finding Eigenvalues and Eigenvectors

Explains the process of finding eigenvalues and eigenvectors for a matrix A by: 1) Forming characteristic polynomial from det(A - λI) and setting = 0 to get characteristic equation, 2) Solving for λ's which are the eigenvalues, 3) Plugging in eigenvalues to A - λI and row reducing to solve for eigenvectors x.

10:08
🧐 Eigenvector Properties and Another Example

Notes eigenvectors represent a form and any scalar multiple cx is also an eigenvector. Demonstrates full process of finding eigenvalues and eigenvectors for a sample 2x2 matrix A. Gets eigenvalues 3 and -1. Shows process of plugging these in to find associated eigenvectors (1,2) and (1,-2) representing vector forms.

15:09
😇 One Final Example with 3x3 Matrix

Goes through detailed example of finding all eigenvalues and eigenvectors for a 3x3 matrix A. Gets eigenvalues 1, -2, and 4. Shows process of plugging these in to solve for associated eigenvector forms (1, 1, -5/3), (0, -2, 1), and (0, 0, 1).

Mindmap
Keywords
💡eigenvalues
Eigenvalues are scalar values associated with a matrix that satisfy the equation Ax = λx, where A is the matrix, x is an eigenvector, and λ is the eigenvalue. Eigenvalues represent special values where the matrix A acts by simple scalar multiplication on the eigenvector x. They allow us to analyze properties of matrices. The video discusses how to find eigenvalues and use them to solve systems of equations and analyze vibrations.
💡eigenvectors
Eigenvectors are vectors that satisfy the equation Ax = λx for a matrix A. They are nonzero vectors that are stretched or shrunk by the matrix A through scalar multiplication. The video shows how to verify if a vector is an eigenvector by checking this equation. Eigenvectors corresponding to each eigenvalue provide insights into the matrix.
💡characteristic polynomial
The characteristic polynomial of a matrix A is found by computing the determinant of A - λI and setting it equal to 0. The roots of this polynomial are the eigenvalues of A. The video demonstrates forming and solving the characteristic polynomial to find eigenvalues.
💡quantum physics
Eigenvalues and eigenvectors have important applications in quantum physics for describing quantum states and energies. The video mentions how these concepts from linear algebra connect to modern physics and quantum theory.
💡matrix
A matrix is a rectangular array of numbers arranged in rows and columns. The video focuses on square matrices, which have the same number of rows and columns. Eigenvalues and eigenvectors are properties of square matrices.
💡row operations
Row operations like adding/subtracting multiples of rows are used to put a matrix in reduced row echelon form. The video uses row operations to solve for eigenvectors by transforming the matrix equation Ax=λx into echelon form.
💡determinant
The determinant of a matrix is a scalar value that provides information about a matrix like its invertibility. The video sets the determinant of A-λI equal to zero to form the characteristic polynomial and find eigenvalues.
💡linear algebra
Linear algebra is the branch of mathematics dealing with linear equations, vector spaces, matrices, and related concepts like eigenvalues. The video illustrates core linear algebra ideas like matrices, vectors, and eigenvalue problems.
💡characteristic equation
The characteristic equation is formed by setting the characteristic polynomial equal to zero and solving for the eigenvalues. The video shows the process of deriving and solving the characteristic equation to determine eigenvalues.
💡natural frequencies
Eigenvalues can represent natural frequencies or resonant frequencies of vibrating systems. The video mentions how eigenvalues relate to analyzing frequencies of mechanical vibrations.
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Transcripts

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