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Q21E

Expert-verifiedFound in: Page 392

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

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.

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

$\pm {\stackrel{\rightharpoonup}{\mathrm{v}}}_{1},\pm {\stackrel{\rightharpoonup}{\mathrm{v}}}_{2},\mathrm{and}\pm {\stackrel{\rightharpoonup}{\mathrm{v}}}_{3}$

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.

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