47. If , then there exist exactly four orthogonal matrices S such that AS is diagonal.
The given statement is FALSE.
Since is symmetric, it has an orthogonal basis consisting of eigenvectors. Let be unit orthogonal eigenvectors of A. Since S is orthogonal diagonal zing matrix of A,S must map to orthogonal eigenvectors.
Thus, we have either and OR and .
This gives a total of 8 possibilities for S.
Therefore, the given statement is FALSE.
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