Eigenvalues and Eigenvectors uniquely define a matrix

In summary, eigenvalues and eigenvectors are scalar values and corresponding vectors used to analyze and understand matrices. They uniquely define a matrix by allowing for diagonalization and easier manipulation. A matrix can only have one set of eigenvalues and eigenvectors, but different matrices can share the same ones. In linear algebra, they help understand matrix behavior and allow for decomposition. In real-world applications, they are used in fields such as physics, engineering, and computer science for various purposes, including image and signal processing and understanding complex systems.
  • #1
zeebo17
41
0
Do a set of Eigenvalues and Eigenvectors uniquely define a matrix since you can produce a matrix [tex]M[/tex] from a matrix of its eigenvectors as columns [tex]P[/tex] and a diagonal matrix of the eigenvalues [tex]E[/tex] through [tex] M=P E P^{\dagger}[/tex]?
 
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  • #2
i'm pretty sure the answer is yes
 
  • #3
It is a bit more complicated. What you say is true if you can diagonalize the matrix. But take a matrix with complex eigenvalues and you are quickly missing eigenvectors. look up algebraic vs. geometric multiplicity of eigenvectors
 
  • #4
Try it with 2 x 2 matrix [0, 1; 0, 0]
 

1. What are eigenvalues and eigenvectors?

Eigenvalues and eigenvectors are mathematical concepts used to analyze and understand matrices. Eigenvalues are scalar values that represent how a linear transformation changes the magnitude of a vector in a specific direction. Eigenvectors are the corresponding vectors that remain in the same direction after the transformation.

2. How do eigenvalues and eigenvectors uniquely define a matrix?

Eigenvalues and eigenvectors form the basis for diagonalizing a matrix, which means expressing the matrix in terms of its eigenvalues and eigenvectors. This unique representation allows for easier manipulation and analysis of the matrix's properties.

3. Can a matrix have more than one set of eigenvalues and eigenvectors?

No, a square matrix can only have one set of eigenvalues and eigenvectors. However, different matrices can have the same eigenvalues and eigenvectors.

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

Eigenvalues and eigenvectors are important in linear algebra because they help us understand the behavior of a matrix under linear transformations. They also allow us to decompose a matrix into its simpler parts, making it easier to work with and solve problems.

5. How are eigenvalues and eigenvectors used in real-world applications?

Eigenvalues and eigenvectors have various applications in fields such as physics, engineering, and computer science. They are used in image and signal processing, data compression, and network analysis, among others. They also play a crucial role in solving differential equations and understanding the behavior of complex systems.

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