Diagonalizable | Generated by AI
To determine which option is necessary for a matrix to be diagonalizable, I’ll examine each statement.
A square matrix A is diagonalizable if and only if there exists an invertible matrix P such that P⁻¹AP is a diagonal matrix. This happens precisely when A has a complete set of linearly independent eigenvectors.
Let’s analyze each option:
a) The matrix must have at least one eigenvalue. While all n×n matrices have at least one eigenvalue (in the complex field), this alone doesn’t guarantee diagonalizability. A matrix could have eigenvalues but still not be diagonalizable if it doesn’t have enough linearly independent eigenvectors.
b) The matrix must have distinct eigenvalues. Having distinct eigenvalues is sufficient but not necessary for diagonalizability. A matrix with repeated eigenvalues can still be diagonalizable if it has enough linearly independent eigenvectors associated with those repeated eigenvalues.
c) The matrix must have enough linearly independent eigenvectors. This is exactly the necessary and sufficient condition for diagonalizability. Specifically, an n×n matrix is diagonalizable if and only if it has n linearly independent eigenvectors, forming a basis for the vector space.
Therefore, option c) is the correct answer: a matrix is diagonalizable if and only if it has enough linearly independent eigenvectors.
I’ll explain why option (c) - having enough linearly independent eigenvectors - is the necessary condition for diagonalizability.
When we say a matrix A is diagonalizable, we mean we can find an invertible matrix P such that P⁻¹AP = D, where D is a diagonal matrix. The columns of P are the eigenvectors of A, and the diagonal entries of D are the corresponding eigenvalues.
Here’s why each option works or doesn’t work:
For option (a): While every square matrix has at least one eigenvalue (in the complex field), having eigenvalues alone doesn’t guarantee diagonalizability. Consider the matrix: [1 1] [0 1]
It has one eigenvalue (λ = 1) with algebraic multiplicity 2, but only one linearly independent eigenvector. This matrix is not diagonalizable despite having eigenvalues.
For option (b): Having distinct eigenvalues is sufficient but not necessary. When a matrix has distinct eigenvalues, the eigenvectors are automatically linearly independent. However, matrices with repeated eigenvalues can still be diagonalizable if they have enough linearly independent eigenvectors.
For option (c): This is precisely the definition of diagonalizability. An n×n matrix is diagonalizable if and only if it has n linearly independent eigenvectors. This is because:
- The diagonalization process requires that P be invertible
- P is invertible if and only if its columns are linearly independent
- The columns of P are the eigenvectors of A
This condition directly addresses the fundamental requirement for diagonalization - having a complete basis of eigenvectors for the vector space.