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We consider $n\times n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two Hermitian matrices $A$ and $B$ are $*-$congruent, then $$(i_+(A), i_-(A), i_0(A))=(i_+(B), i_-(B), i_0(B)).\qquad{(1)}$$ If two general matrices $A$ and $B$ are $*-$congruent, (1) may not hold. But I have come up a counterexample for this. Moreover, whether a matrix and its transpose are always $*-$congruent?

We consider $n\times n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two Hermitian matrices $A$ and $B$ are $*-$congruent, then $$(i_+(A), i_-(A), i_0(A))=(i_+(B), i_-(B), i_0(B)).\qquad{(1)}$$ If two general matrices $A$ and $B$ are $*-$congruent, (1) may not hold (can you provide an example?). Moreover, whether a matrix and its transpose are always $*-$congruent?

We consider $n\times n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two Hermitian matrices $A$ and $B$ are $*-$congruent, then $$(i_+(A), i_-(A), i_0(A))=(i_+(B), i_-(B), i_0(B)).\eqno(1)$$ If two general matrices $A$ and $B$ are $*-$congruent, (1) may not hold. But I have come up a counterexample for this. Moreover, whether a matrix and its transpose are always $*-$congruent?

We consider $n\times n$ complex matrices. Let $i_+(A), i_-(A), i_0(A)$ be the number of eigenvalues of $A$ with positive real part, negative real part and pure imaginary. It is well known if two Hermitian matrices $A$ and $B$ are $*-$congruent, then $$(i_+(A), i_-(A), i_0(A))=(i_+(B), i_-(B), i_0(B)).\qquad{(1)}$$ If two general matrices $A$ and $B$ are $*-$congruent, (1) may not hold. But I have come up a counterexample for this. Moreover, whether a matrix and its transpose are always $*-$congruent?