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 hermitian_positive_semi-definite_matrix [2013/09/19 20:50]nikolaj hermitian_positive_semi-definite_matrix [2014/03/21 11:11] (current) Both sides previous revision Previous revision 2013/09/19 20:50 nikolaj 2013/09/19 20:50 nikolaj 2013/09/19 20:49 nikolaj 2013/09/19 20:48 nikolaj 2013/09/19 20:48 nikolaj created Next revision Previous revision 2013/09/19 20:50 nikolaj 2013/09/19 20:50 nikolaj 2013/09/19 20:49 nikolaj 2013/09/19 20:48 nikolaj 2013/09/19 20:48 nikolaj created Line 1: Line 1: ===== Hermitian positive semi-definite matrix ===== ===== Hermitian positive semi-definite matrix ===== - ==== Definition ​==== + ==== Set ==== - | @#88DDEE: $n\in\mathbb N$ | + | @#55CCEE: context ​    | @#55CCEE: $n\in\mathbb N$ | - | @#FFBB00: $A \in \mathrm{it}(n)$ | + | @#FFBB00: definiendum ​| @#FFBB00: $A \in \mathrm{it}(n)$ | - | @#55EE55: $A \in \mathrm{HermitianMatrix}(n)$ | + | @#55EE55: postulate ​  | @#55EE55: $A \in \mathrm{HermitianMatrix}(n)$ | | $x \in \text{ColumnVector}(n,​\mathbb C)$ | | $x \in \text{ColumnVector}(n,​\mathbb C)$ | - | @#55EE55: $A \in \mathrm{SquareMatrix}(n,​\mathbb C)$ | + | @#55EE55: postulate ​  | @#55EE55: $A \in \mathrm{SquareMatrix}(n,​\mathbb C)$ | - | @#55EE55: $x^* A\ x \ge 0$ | + | @#55EE55: postulate ​  | @#55EE55: $x^* A\ x \ge 0$ | ==== Discussion ==== ==== Discussion ==== - If $B$ is real/​complex,​ then $C=B^*B$ is Hermitian positive semi-definite. === Reference === === Reference === Wikipedia: [[http://​en.wikipedia.org/​wiki/​Positive-definite_matrix|Positive-definite matrix]] Wikipedia: [[http://​en.wikipedia.org/​wiki/​Positive-definite_matrix|Positive-definite matrix]]