Math Problem Statement

Find the eigen value and eigen vectors a. A = [1 1 0; 0 1 1]

Solution

To find the eigenvalues and eigenvectors of matrix AA, we follow these steps:

Matrix:

1 & 1 & 0 \\ 0 & 1 & 1 \end{bmatrix}$$ Since this is a $$2 \times 3$$ matrix (rectangular), it doesn't have the same set of eigenvalues and eigenvectors as a square matrix would. Eigenvalues and eigenvectors are typically defined for square matrices because they relate to how a matrix can be factored into components based on its characteristic polynomial, and for that, we need a square matrix. Would you like to find another type of factorization for this matrix, or would you prefer an explanation about why eigenvalues and eigenvectors are only well-defined for square matrices? Here are some follow-up questions to expand on this: 1. Would you like to calculate the singular value decomposition (SVD) of this rectangular matrix? 2. Should we try another square matrix example for eigenvalue/eigenvector calculation? 3. Would you like a discussion on matrix transformations in general? 4. Should we explore how non-square matrices can be used in systems of equations? 5. Are you interested in learning how rectangular matrices are used in applications such as machine learning? **Tip:** Eigenvalues and eigenvectors are powerful tools for understanding matrix behavior, but they are only defined for square matrices.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Eigenvectors
Matrix Theory

Formulas

Ax = λx (Eigenvalue equation)
det(A - λI) = 0 (Characteristic equation)

Theorems

Eigenvalue Theorem
Spectral Theorem (for square matrices)

Suitable Grade Level

Undergraduate