Math Problem Statement

True or False? If the eigenvalues of a matrix are not distinct, then the matrix is not diagonalizable.

Solution

The question you're working on asks:

"True or False? If the eigenvalues of a matrix are not distinct, then the matrix is not diagonalizable."

The statement is False.

Explanation: A matrix can still be diagonalizable even if it has repeated (non-distinct) eigenvalues. What matters is whether there are enough linearly independent eigenvectors to form a basis for the space. If the geometric multiplicity of each eigenvalue (the number of linearly independent eigenvectors corresponding to it) equals its algebraic multiplicity (the number of times the eigenvalue appears in the characteristic equation), then the matrix is diagonalizable, even if the eigenvalues are not distinct.

Would you like further details on this concept or any clarification?

Here are five related questions to consider:

  1. What conditions are necessary for a matrix to be diagonalizable?
  2. How can you determine the eigenvalues and eigenvectors of a matrix?
  3. What is the difference between algebraic and geometric multiplicity of eigenvalues?
  4. Can a defective matrix (one that is not diagonalizable) be transformed in any way to simplify analysis?
  5. How does the Jordan canonical form relate to non-diagonalizable matrices?

Tip: Always check both the algebraic and geometric multiplicities of eigenvalues when determining if a matrix is diagonalizable.

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Math Problem Analysis

Mathematical Concepts

Linear Algebra
Eigenvalues
Eigenvectors
Matrix Diagonalization

Formulas

Characteristic equation: det(A - λI) = 0
Algebraic Multiplicity = number of times an eigenvalue appears
Geometric Multiplicity = number of independent eigenvectors

Theorems

Diagonalization Theorem

Suitable Grade Level

Undergraduate