Geodesic
On the multiplicities of eigenvalues of a pair of quadratic forms
Matematičeskoe obrazovanie, Tome 111 (2024) no. 3, pp. 26-30

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We consider a pair of real quadratic forms, one of which is positive definite. It is proven that it is possible without calculating the eigenvalues of this pair of forms, to find out if any of them are multiples. One can also express the sum of the squares of the multiplicities of these numbers through the rank of some auxiliary matrix. 
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     author = {N. V. Ilyushechkin},
     title = {On the multiplicities of eigenvalues of a pair of quadratic forms},
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N. V. Ilyushechkin. On the multiplicities of eigenvalues of a pair of quadratic forms. Matematičeskoe obrazovanie, Tome 111 (2024) no. 3, pp. 26-30. http://geodesic.mathdoc.fr/item/MO_2024_111_3_a3/