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In the answer of to the question Characterization of positive definite matrix with principal minors I saw such assertion: $Ak=LkDkL∗k \implies Ak$ is congruent to Dk . Tell me please, why is it right? Matrices are similar when $A=LDL^-1$, not $L*$, isn't it? And, regarding to another answer, why "if hermitian matrix A is not positive definite, it must possess at least two(???) negative eigenvalues"? Why not only 1? It has some connection with complex field?

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