Question:
The necessary condition to diagonalize a matrix is that
The necessary condition to diagonalize a matrix is that
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Distinct eigenvalues are a "bonus," but independent eigenvectors are a "must-have" for diagonalization.
Updated On: Jul 3, 2026
- the matrix is non-singular
- its eigen vectors should be independent
- its eigen values should be real
- its all eigen values should be distinct
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The Correct Option is B
Solution and Explanation
Step 1: Concept
For an $n \times n$ matrix $A$ to be diagonalizable, there must exist an invertible matrix $P$ such that $P^{-1}AP = D$.
Step 2: Meaning
The columns of matrix $P$ are the eigenvectors of $A$. For $P$ to be invertible, its columns must be linearly independent.
Step 3: Analysis
While having distinct eigenvalues (D) is a *sufficient* condition, it is not *necessary* (e.g., the Identity matrix is diagonalizable but has repeated eigenvalues). However, having $n$ linearly independent eigenvectors is a requirement.
Step 4: Conclusion
Thus, the independence of eigenvectors is the fundamental necessary condition. Final Answer: (B)
For an $n \times n$ matrix $A$ to be diagonalizable, there must exist an invertible matrix $P$ such that $P^{-1}AP = D$.
Step 2: Meaning
The columns of matrix $P$ are the eigenvectors of $A$. For $P$ to be invertible, its columns must be linearly independent.
Step 3: Analysis
While having distinct eigenvalues (D) is a *sufficient* condition, it is not *necessary* (e.g., the Identity matrix is diagonalizable but has repeated eigenvalues). However, having $n$ linearly independent eigenvectors is a requirement.
Step 4: Conclusion
Thus, the independence of eigenvectors is the fundamental necessary condition. Final Answer: (B)
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