📚 Topic Summary
A symmetric matrix is a square matrix that is equal to its transpose. In other words, if $A$ is a symmetric matrix, then $A = A^T$. A key property of real symmetric matrices is that their eigenvalues are always real numbers, and eigenvectors corresponding to distinct eigenvalues are orthogonal. This makes them particularly useful in many applications of linear algebra, especially in physics and engineering.
This worksheet will guide you through understanding these properties and applying them to find eigenvalues of symmetric matrices.
🧠 Part A: Vocabulary
Match the term with its correct definition:
- Term: Eigenvalue
- Term: Eigenvector
- Term: Symmetric Matrix
- Term: Transpose
- Term: Orthogonal
- Definition: A vector that, when multiplied by a matrix, results in a scalar multiple of itself.
- Definition: A scalar $\lambda$ such that $A\mathbf{v} = \lambda\mathbf{v}$ for some nonzero vector $\mathbf{v}$.
- Definition: A matrix $A$ such that $A = A^T$.
- Definition: Two vectors whose dot product is zero.
- Definition: A matrix formed by interchanging the rows and columns of a given matrix.
Instructions: Write the number of the correct definition next to each term. For example, if you think the definition of 'Eigenvalue' is #3, write '3' next to 'Eigenvalue'.
✍️ Part B: Fill in the Blanks
Complete the following paragraph with the correct words:
A __________ matrix is equal to its __________. The __________ of a symmetric matrix are always __________. Eigenvectors corresponding to different eigenvalues are __________. Therefore, symmetric matrices are very important in __________ and __________.
Word Bank: orthogonal, transpose, eigenvalues, physics, symmetric, real, engineering
🤔 Part C: Critical Thinking
Explain why the eigenvalues of a real symmetric matrix are always real. What implications does this have for applications in fields like quantum mechanics or structural engineering?