Question

正定矩阵A的对角线元素之和大于0．若矩阵A对角线元素全大于0，则其正定？

A.

Answer 1

【答案】： style='WHITE-SPACE: normal; TEXT-TRANSFORM: none; WORD-SPACING: 0px; COLOR: rgb(51,51,51); FONT: 14px/28px zuoyeFont_mathFont, 'Microsoft Yahei', 宋体, sans-serif; WIDOWS: 1; LETTER-SPACING: normal; BACKGROUND-COLOR: rgb(255,255,238); TEXT-INDENT: 0px; -webkit-text-stroke-width: 0px'>取 x = (0,...,1,...,0)^T,第i个分量为1,其余为0
则 x^TAx = aii > 0.
即得A的主对角线上元素都大于0. style='WHITE-SPACE: normal; TEXT-TRANSFORM: none; WORD-SPACING: 0px; FLOAT: none; COLOR: rgb(51,51,51); FONT: 14px/28px zuoyeFont_mathFont, 'Microsoft Yahei', 宋体, sans-serif; WIDOWS: 1; DISPLAY: inline !important; LETTER-SPACING: normal; BACKGROUND-COLOR: rgb(255,255,238); TEXT-INDENT: 0px; -webkit-text-stroke-width: 0px'>
则A正定