SOLUTION: Hi, i don't really know what to do here. The linear operator A is called the idempotent or the projection operator if A^2 = A. What is the canonical operator of the idempotent o

Question 1194124: Hi, i don't really know what to do here.
The linear operator A is called the idempotent or the projection operator if A^2 = A. What is the canonical operator of the idempotent operator Jordan?
in shape?
Answer by parmen(42) (Show Source): You can put this solution on YOUR website!
**Key Properties of Idempotent Operators:**
* **Definition:** An operator A is idempotent if A² = A.
* **Eigenvalues:** The only possible eigenvalues of an idempotent operator are 0 and 1.
* **Diagonalizability:** Idempotent operators are always diagonalizable.
**Canonical Form of an Idempotent Operator:**
The canonical form of an idempotent operator is a diagonal matrix with only 0s and 1s on the diagonal.
**Explanation:**
* **Diagonalization:** Since idempotent operators are diagonalizable, there exists an invertible matrix P such that:
P⁻¹AP = D
where D is a diagonal matrix.
* **Eigenvalues and Diagonal:**
- The diagonal entries of D represent the eigenvalues of A.
- Since the only eigenvalues of an idempotent operator are 0 and 1, the diagonal of D will consist of only 0s and 1s.
**Example:**
Let A be an idempotent operator. Its canonical form (Jordan form) would be:
```
D =
| 1 0 0 0 |
| 0 1 0 0 |
| 0 0 0 0 |
| 0 0 0 0 |
```
This represents a 4x4 matrix with two eigenvalues: 1 (with multiplicity 2) and 0 (with multiplicity 2).
**In Summary:**
The canonical form of an idempotent operator is a diagonal matrix with only 0s and 1s on the diagonal, reflecting its eigenvalues and its diagonalizability.

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