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Expert-verified Found in: Page 392 ### Linear Algebra With Applications

Book edition 5th
Author(s) Otto Bretscher
Pages 442 pages
ISBN 9780321796974 # Consider a symmetric 3x3 matrix A with eigenvalues 1,2 and 3 how many different orthogonal matrices s are there such that ${{\mathbf{S}}}^{\mathbf{-}\mathbf{1}}{\mathbf{AS}}$is diagonal?

Thus, the different orthogonal matrices are 48

See the step by step solution

## Step 1: The Orthogonal Matrix

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 ${\mathrm{Q}}^{\mathrm{T}}$ is the transpose of Q and l is the identity matrix is one approach to describe this.

## Step 2: Determine the orthogonal Matrices

Since A has three distinct eigenvalues, then we have possible distinct orthonormal eigenvectors corresponding to these eigenvalues:

$±{\stackrel{⇀}{\mathrm{v}}}_{1},±{\stackrel{⇀}{\mathrm{v}}}_{2},\mathrm{and}±{\stackrel{⇀}{\mathrm{v}}}_{3}$

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.

Therefore 6x4x2=48

The different orthogonal matrices are 48. ### Want to see more solutions like these? 