What happens if matrix A*A* was not diagonalizable?

If a matrix \( A \) is not diagonalizable, it means that \( A \) cannot be expressed as \( PDP^{-1} \), where \( D \) is a diagonal matrix and \( P \) is an invertible matrix composed of the eigenvectors of \( A \).

### What Happens if \( A \) is Not Diagonalizable?

1. **No Simple Diagonal Representation**: 
   - If \( A \) is not diagonalizable, we cannot find a basis of eigenvectors that would allow us to write \( A \) as a diagonal matrix \( D \) under a similarity transformation. As a result, computing powers of \( A \) directly using diagonalization is not possible.

2. **Jordan Canonical Form**:
   - For matrices that are not diagonalizable, we can use the Jordan canonical form (Jordan normal form) as an alternative. The Jordan form of a matrix \( A \), denoted as \( J \), is almost diagonal but contains ones on the superdiagonal (just above the main diagonal) to account for the generalized eigenvectors.
   - \( A \) can be expressed as \( A = PJP^{-1} \), where \( J \) is the Jordan form of \( A \), and \( P \) is a matrix whose columns are the generalized eigenvectors of \( A \).

3. **Calculating Powers with Jordan Form**:
   - If \( A \) is in Jordan form \( J \), then \( A^n = P J^n P^{-1} \). The matrix \( J^n \) is easier to compute because raising a Jordan block to a power involves known formulas that mix diagonal elements and their corresponding superdiagonal entries.
   - For example, if \( J \) has a Jordan block \( J_k \) of size \( k \) corresponding to an eigenvalue \( \lambda \), then \( J_k^n \) has \( \lambda^n \) along the diagonal and involves binomial coefficients on the superdiagonal.

4. **More Complex Computation**:
   - Even though powers of \( A \) can still be computed using the Jordan form, the calculation becomes more complex than for diagonalizable matrices. The complexity arises from the need to manage off-diagonal elements that are introduced by the Jordan form's structure.

5. **Practical Implications**:
   - In practical terms, if \( A \) is not diagonalizable, the process of finding \( A^n \) involves more steps, and the resulting expressions are often more complicated due to the interaction between eigenvalues and generalized eigenvectors.
   - Certain theoretical tools, like functions of matrices (e.g., matrix exponentials), also become more intricate to define and compute when a matrix is not diagonalizable.

### Conclusion

If a matrix \( A \) is not diagonalizable, one must use the Jordan form to compute higher powers or other functions of \( A \). While this method is more complex than using diagonalization, it still provides a structured way to handle non-diagonalizable matrices.

Would you like to delve deeper into the Jordan form or explore another related topic?

### Further Questions

1. How do you find the Jordan canonical form of a matrix?
2. What are generalized eigenvectors, and how do they differ from regular eigenvectors?
3. Can all matrices be put into Jordan form, and under what conditions?
4. How does the Jordan form help in computing matrix exponentials?
5. What are some practical applications of the Jordan canonical form in differential equations or other areas?

### Tip

When dealing with non-diagonalizable matrices, always check if the matrix can be simplified using the Jordan canonical form. This approach, while more involved, can still provide a systematic way to analyze matrix behavior.

Explore what it means when a matrix A is not diagonalizable and how the Jordan canonical form provides an alternative representation. Learn about the implications for matrix computations, including calculating higher powers and matrix exponentials.

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