The properties of eigenvalues are fundamental concepts in linear algebra, especially when dealing with matrices. Here are 10 important properties of eigenvalues:

### 1. **Eigenvalues of Diagonal Matrices** 
   - For a diagonal matrix \( D \), the eigenvalues are simply the entries on the diagonal. For example, if \( D = \text{diag}(d_1, d_2, \ldots, d_n) \), then the eigenvalues are \( d_1, d_2, \ldots, d_n \).

### 2. **Trace and Sum of Eigenvalues**
   - The trace of a square matrix \( A \), denoted as \( \text{tr}(A) \), is equal to the sum of its eigenvalues. Mathematically, if \( \lambda_1, \lambda_2, \ldots, \lambda_n \) are the eigenvalues of \( A \), then:
     \[
     \text{tr}(A) = \sum_{i=1}^{n} \lambda_i
     \]

### 3. **Determinant and Product of Eigenvalues**
   - The determinant of a square matrix \( A \), denoted as \( \det(A) \), is equal to the product of its eigenvalues. That is:
     \[
     \det(A) = \lambda_1 \cdot \lambda_2 \cdot \ldots \cdot \lambda_n
     \]

### 4. **Eigenvalues of Triangular Matrices**
   - For triangular matrices (upper or lower), the eigenvalues are the elements on the main diagonal. For example, if \( A \) is a triangular matrix with diagonal entries \( a_{11}, a_{22}, \ldots, a_{nn} \), these entries are its eigenvalues.

### 5. **Eigenvalues of Inverse Matrices**
   - If \( \lambda \) is an eigenvalue of a matrix \( A \), then \( \frac{1}{\lambda} \) is an eigenvalue of \( A^{-1} \), provided \( A \) is invertible.

### 6. **Eigenvalues of Transposed Matrices**
   - The eigenvalues of a matrix \( A \) and its transpose \( A^T \) are identical. This means that if \( \lambda \) is an eigenvalue of \( A \), then it is also an eigenvalue of \( A^T \).

### 7. **Eigenvalues of Similar Matrices**
   - Similar matrices share the same eigenvalues. If two matrices \( A \) and \( B \) are similar, meaning \( B = P^{-1}AP \) for some invertible matrix \( P \), then \( A \) and \( B \) have the same eigenvalues.

### 8. **Sum of Powers of Eigenvalues (Cayley-Hamilton Theorem)**
   - A matrix satisfies its own characteristic equation. If the characteristic polynomial of a matrix \( A \) is \( p(\lambda) = \det(\lambda I - A) = 0 \), then substituting \( A \) itself into this polynomial yields \( p(A) = 0 \). Hence, the eigenvalues satisfy this polynomial equation.

### 9. **Multiplicative Properties with Scalars**
   - If \( A \) is a matrix and \( c \) is a scalar, then the eigenvalues of the matrix \( cA \) are \( c\lambda_1, c\lambda_2, \ldots, c\lambda_n \), where \( \lambda_1, \lambda_2, \ldots, \lambda_n \) are the eigenvalues of \( A \).

### 10. **Complex Conjugate Pairs for Real Matrices**
   - If \( A \) is a real matrix and has complex eigenvalues, then those eigenvalues occur in complex conjugate pairs. For example, if \( a + bi \) is an eigenvalue of a real matrix \( A \), then \( a - bi \) is also an eigenvalue.

Would you like more details on any specific property or have any questions?

### Related Questions
1. How are eigenvalues used to determine matrix invertibility?
2. What is the significance of eigenvectors associated with eigenvalues?
3. How does the determinant change if an eigenvalue is zero?
4. How do eigenvalues relate to stability in systems of differential equations?
5. How can eigenvalues help in simplifying matrix computations?

### Tip
The trace and determinant properties of eigenvalues make it easy to check certain aspects of a matrix without full diagonalization!

Explore 10 key properties of eigenvalues, covering fundamental concepts in linear algebra. This article discusses eigenvalue properties such as trace, determinant, diagonal and triangular matrices, as well as the Cayley-Hamilton theorem. Perfect for undergraduate-level students.

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