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?
Asked
Active
Viewed 202 times
0
-
1Welcome to Math.SE! Please use MathJax. For some basic information about writing math at this site see e.g. basic help on mathjax notation, mathjax tutorial and quick reference, main meta site math tutorial and equation editing how-to. – GNUSupporter 8964民主女神 地下教會 Mar 02 '19 at 01:53
-
2Matrices $A$ and $B$ are congruent if $B = P A P^$ for some invertible matrix $P$ (often $P^$ in this definition is replaced with $P^{\top}$; the two definitions coincide for real matrices). Matrices $A$ and $B$ are similar if $B = P A P^{-1}$ for some $P$. The second claim is not true; it fails for the zero matrix, which is Hermitian and not positive definite. – Travis Willse Mar 02 '19 at 03:09