放映：《矩阵论及其应用》第3章§1向量范数与矩阵范数

1、, VV , | 0 0 | , 0 |0l) ( , | |Rkkklkl12k | |)(FV ,V|, 0 |00 |)( | |Fkk| |V). | , (VV, VV , | V| | | | | | | | | | | | | | | V,| |)( | )(VfV ,00 |0,00.lim0)(f0 | | )()(|00ff0niixx1 1|nTnCxxxx 21niixx1 2 2|)1 (| )(11 pxxpnipip|max 1inixpxxx| ,| ,|21pnipipx1)(1 |lim| xpnipix1)(1 |pnipix10)(1 | 0ixn1p,

2、 |max | 10iniixx| 0ix|0ix)(p |max| 1inixxniixx1 1|nTnCxxxx 21niixx1 2 2|)1 (| )(11 pxxpnipippxxx| ,| ,|21niix1 2 2|.nVnV12,n , nV2|nVnTnCxxxx 211122 nnxxx nVnCnnCV nVnC| ,| ,|1p01211 1nnV nni1 11|nni 1 2 21|1 |)(厘米厘米 5)(毫米毫米 50 ,)(米米 50.0 ,)(公里公里 50.0000 ,babMm| | |nVba| , |Mm 0,nV a|b|.| |baaa| |ab

3、ba| | | |cacbba| | | | ,| |”“ nVba| , | ),(21nxxxnV| ,21neeenV,2211nnnVexexex, 2211nnnVeyeyey | | ),( ),( 2121nnyyyxxx | |)(| 1 niiiieyx| 1 niiiieyx), 2 , 1,( 0nixyiibabMm| | | 2122| / | (,) | / |aanbbmyyyM nVba| , |banxxx|),(21 1| ),( 2222121nnyyyyyyS. 0MS, 0m)0( 2211nnnVexexex0nCSyyyeyxnniiinii),(

4、 , |211 1 2niix1 1|nTnCxxx 21nC| ,| ,|21|max1inixn|n|max|1inix| |1 1nniix1 222|21|maxinixn2|n21|maxinix2| | |1 2n.| |1.| |221| |121| |1nn, nmCA| AAAmnC minjijaA1 1 1| |max|11ijnjmiaA minjijaA1 1 22| |mnmmnnaaaaaaaaaA222221112111mnmnaaaaA1111mnnmaaaaA1111, nmCA| AAAmnC minjijaA1 1 1| |max|11ijnjmiaAC

5、ba , | |baab111 1 , 111 1BA2 |AB2222 AB| 1BA| BAAB| | |baAxAx | , | , |pmpnnmCCC,| | |BAABpnnmCBCA, ( ,| | ) , ( ,| | )nmabCC | nmC| nnmCxCA, minjijaA1122| |max |,ijjiaAnmijCaA 2| |AAF)( | |nabCxxAAxba| ,|mnCC , nmCAba| ,|,| A)0,( | | xCxAxxAnba1|,|aaxx) 1| , ( | |anbCAAbAf| )( | AbAa|max1| 1 | , an

6、CSbSA|max|AA|max |1|AxAx 0|max ()|nbxaAxxCx OA 0 |A0 |A0 |max1|Axx) 1 ,( 0 |xCxAxn)0,( 0 |xCxxxAn)0,( 0 |xCxAxn)0,( 0 xCxAxnOA |max |1|AxAx )( | 00nCxAxOA 0 |A| kA|max| 1|Axkx | Ak Ck |max1|kAxx |max |1|AxAx )( | 00nCxAxOA 0 |A| kA| Ak Ck | |ABAB |max |1|AxAx )( | 00nCxAxOA 0 |A| kA| Ak Ck |BA| BA |

7、AB | | |AB 12mnCC ,| , , 21nmC11|1|max| 1 AxAx21|2|max| 2AxAx|max| 1|AxAxmiijnjaA111|max| 1 ）（njijmiaA11|max| 2 ）（112 ,| 3A）（A AH,nmCA）（ 1miijnjmiijnjaAaA111111|max|max|miijnjaA111|max| 1 ）（njijmiaA11|max| 2 ）（112 ,| 3A）（A AH,nmCA）（ 1miijnjmiijnjaAaA111111|max|max|6 312 ,131 max| 1A 3 , 113, 3 max|

8、A113 HA A30123213120231 ii14252514ii167282HA A29142, 12914 |2A 14252514 ii, 3012321 iA.| ,| ,|21AAA )( | max )(pSA | )(AASp)(ASpA)(AAxyO)(A) ( | 2HAAAA)( | 2AAAAATH,nnCAAACAnn | , 00 xE() max | () pASA )(AAxyO)(A| )(AA |0000 |xx|0 Ax|0 xA | | 0A| )( AA 00 , )( xASp|A,nnCA | )(AASp)(ASpA6 max1 , 2 , 61|A max3 , 3 , 3|A3 1) 1()3( 121 110 03)(Af 231,2313,)( iiASp, 01211100 3A)( , )(,|1AASAAp) 1)(3(23 | | 231 ,231 3 ,max)(iiAmax 1 , 1 , 3 xAxACAnn ,A| | 11AAxA xxA110 11A| )( |AA |111 )( |A