Consider a symmetric 3x3 matrix A with eigenvalues 1,2 and 3 how many different orthogonal matrices s are there such that is diagonal?
Thus, the different orthogonal matrices are 48
An orthogonal matrix, also known as an orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors in linear algebra.
Where is the transpose of Q and l is the identity matrix is one approach to describe this.
Since A has three distinct eigenvalues, then we have possible distinct orthonormal eigenvectors corresponding to these eigenvalues:
Now, when obtaining the orthogonal matrix S, the first column would have 6 choices, moving to the second column we will have 4 choices (since we can't have a corresponding eigenvector to the same eigenvalue used in the first column), and finally moving to the third column we will have 2 choices left.
The different orthogonal matrices are 48.
94% of StudySmarter users get better grades.Sign up for free