7 The spectral theorem and Hadamard’s determinant problem

Lemma 7.1

If \(A\) is a real symmetric \(n \times n\) matrix that is not diagonal, that is \(\operatorname {Od}(A) {\gt} 0\), then there exists \(U \in O(n)\) such that \(\operatorname {Od}(U^TAU){\lt}\operatorname {Od}(A)\).

Proof ▶

TODO

Theorem 7.2

For every real symmetric matrix \(A\) there is a real orthogonal matrix \(Q\) such that \(Q^{T}AQ\) is diagonal.

Proof ▶

TODO

Theorem 7.3

There exists an \(n \times n\) matrix with entries \(\pm 1\) whose determinant is greater than \(\sqrt{n!}\).

Proof ▶

TODO