HW12.4. Diagonalize a symmetric 2×2 matrix Consider a 2 x 2 matrix A = . Find an orthogonal matrix 2 x 2-matrix Q and a diagonal 2 x 2- matrix D such that A = QDQT. Q = D= Note: In order to be accepted as correct, all entries of the matrices A – QDQT and QTQ – I must have absolute value smaller than 0.05. Save & Grade 5 tries left Save only
The Correct Answer and Explanation is :
To diagonalize a symmetric (2 \times 2) matrix ( A ), we follow these steps:
- Find the Eigenvalues: Solve ( \det(A – \lambda I) = 0 ) to find the eigenvalues ( \lambda_1 ) and ( \lambda_2 ).
- Find the Eigenvectors: Solve ( (A – \lambda I)x = 0 ) for each eigenvalue to get eigenvectors.
- Construct the Orthogonal Matrix ( Q ): Normalize the eigenvectors and arrange them as columns in ( Q ).
- Construct the Diagonal Matrix ( D ): Place the eigenvalues as diagonal entries.
- Verify ( A = QD Q^T ): Check the reconstruction with a tolerance of 0.05.
- Ensure ( Q ) is Orthogonal: Verify ( Q^T Q = I ).
Now, for the explanation:
Explanation:
A symmetric matrix is always diagonalizable with an orthogonal matrix of eigenvectors. The process starts by computing the characteristic equation ( \det(A – \lambda I) = 0 ), yielding two real eigenvalues. Next, we determine corresponding eigenvectors and normalize them to form an orthogonal matrix ( Q ). The diagonal matrix ( D ) contains the eigenvalues. The relationship ( A = QD Q^T ) ensures the correctness of our diagonalization.
Orthogonal matrices have the property ( Q^T Q = I ), meaning they preserve angles and lengths. This is crucial in physics and engineering applications, such as quantum mechanics and principal component analysis.
To validate correctness, we compute ( A – QD Q^T ) and ( Q^T Q – I ) and ensure all entries are within a small tolerance (0.05). This confirms numerical accuracy.
Diagonalization simplifies matrix operations, making exponentiation and function computation more efficient. For example, powers of ( A ) can be computed as ( A^n = Q D^n Q^T ), where ( D^n ) is easy to compute. This is particularly useful in solving differential equations and modeling dynamical systems.
Thus, diagonalizing symmetric matrices is fundamental in linear algebra and has widespread applications in applied mathematics, physics, and computer science.