&du i b P aui) 2aU2) a WW盏 a(能 u aqui csthn2trint ∈Rl 66 for an≠,( n is SPD) arAxi Matrix Properties We give below the definition of some matrix classes and their main properties w sthn2trin Positiv2 D2finit2 (SPDt We sa that a matrix A sitive definite if M> for an non-zero vector For s mmetric matrices this condition is e Divalent to rehiring that all the elg es of the matrix be positive To show this we note that if n is s, mmetric and has real coet cients, it can bwwwritten as M=e6e, where 6 is the diagonal matrix of eigenvalues and e is an orthonormal transformation Iwe_A sinve e is non-Singtlar), impies br>, for an -#, or an. all the greater than zero bviousl, an matrix which is i s b is also non-sing ular and therefore invertiblE Ci=e 6 ele t Dieronel Dominent ft atv W that a matrix M diagonal dominant ifå✷æ❃ç❏è✲é▼ê ë★ì í î❨ï✷ð ñò ò ò ò ò ò òó ô õ í ö ÷❄÷t÷ ö õ í ô õ í ø ø ø ø ø ø ö ø ø ø ø ø ø ø ø ø ö ø ø ø ø ø ø õ í ô õ í ö ÷t÷t÷ ö õ í ô ùtúú ú ú ú ú ú ûýü ÿþ ì ñò ò ò ò òó ÿ✁￾ þÿþ ð ø ø ÿ✄✂✆☎ þ ø ￾ ÿ✄✂þ ùtúú ú ú ú û ü ✝ ì ñò ò ò ò òó ✝✟✞ï ￾✡✠ ✝✟✞ï ð ✠ ø ø ✝✟✞ï ✂✆☎ ø ￾ ✠ ✝✟✞ï ✂☛✠ ùtúú ú ú ú û ☞✍✌✏✎✁✑✒✑✔✓✖✕✘✗✚✙✜✛✆✢ ë✤✣✦✥✧✂✩★✪✂ ÿþ ü ✝ ✣✦✥✧✂ ✫✁✬✜✭✩✬✯✮ ✌✱✰✳✲✵✴✱✕✶✙✜✰✩✷ å✷æ❃ç❏è✲é✹✸ ✺✘✻ ë ✷✼✰✩✷✱✽ ✻ ✙✜✷✿✾✳✴✿✲✜❀✪✗❂❁ ❃❅❄❇❆❉❈❋❊✆● ❍ ì❏■ ❍ ￾ ü ❍ ð ü ❑✘❑✡❑ ü ❍ ✂✩▲✶▼ ❍▼ ë ❍ ì í î✍ïð ✞❍ ð ￾❖◆ ✂ P◗❘ ð ✞❍ ◗ õ ❍ ◗ ☎ ￾✘✠ ð ◆ ❍ ð ✂ ✠ ❙❉❚✡❊☛❯❱❚ ❍▼ ë ❍ ❲ ö❅❳✆❨❄❇❆❉❈❋❊✆●❩❍ ❭❬ ö ✞ë❫❪❵❴ ✌✱❛❝❜✠ ❞❢❡ ë ÿþ ì ✝ ❣ ÿþ ✓✖❤✁✙ ✻✕ ✻✐❈❥❊☛❦ ❪❵❴ ✴✿✷✿✙✜❧✏✴✿✓ ❞ í❄ö ♠♦♥✏♣rqts ✉✇✈✏♣②①④③⑥⑤⑧⑦❝①❇♥❥⑨❢q✚①✆♣⑩③❶q❋❷ ❸❖❹❅❚✐❺❻❈✚❼⑩❆ ❪✯❽✚ë ❹☛❈❴ ❈❾❊✖❿☛❺❾➀✩❚✘❆❖❄❨✱➁❆⑩❄➁ ❚✡❆②❼ ❪❚ ❴ ❼⑩❹✳❈✚❼❖➂❪✯➃❵➃ ➀✳❚➄❚❽ ➁ ➃❄❪❼⑩❚✶❦ ❪❊❩❼⑩❹☛❚✐❈❥❊☛❈➃●❴②❪➅❴ ➆♦❚❝➇ ø ❪❵➈❚✐➀✳❚➃❄✚➂➉❼⑩❹❅❚➊❦❅❚❱➋☛❊❪❼ ❪❄❥❊✦❄❨ ❴❄❥❺➌❚➄❺➌❈❋❼⑩❆ ❪➍❽ ❯➃❈❴✍❴❚ ❴ ❈❥❊☛❦❂❼⑩❹❅❚❪❆➎❺❻❈❪❊ ➁ ❆✍❄➁ ❚✘❆⑩❼ ❪❚❴ ø ✌✁✎✁✑✔✑✔✓✖✕✘✗✚✙✜✛➏❛➐✰✻ ✙✜✕✡✙⑥➑✩✓➒❜➓✓❇➔✟✷✿✙⑥✕✡✓✔☞✍✌✁❛❝❜→✢ ➆♦❚ ❴❈④●✔❼⑩❹☛❈❋❼❂❈➣❺❻❈✚❼⑩❆ ❪✯❽⑧↔ ❪❵❴ ➁ ❄❴⑩❪❼ ❪❵➈❚➒❦✪❚✘➋☛❊❪❼⑩❚ ❪❨ ❍▼↔❍ ❲ ö♦❨❄❥❆➏❈❥❊✆●✒❊❅❄❇❊✪↕➛➙✡❚✘❆✍❄ ➈❚✶❯r❼✍❄❥❆➎❍ ø ❃☛❄❥❆ ❴●✆❺➌❺➜❚✘❼⑩❆ ❪❯➄❺❻❈❋❼⑩❆ ❪❯✘❚❴ ❼✍❹❪❵❴ ❯❱❄❇❊☛❦❪❼ ❪❄❥❊ ❪➅❴ ❚✡➝✖❿❪❵➈❈➃❚✘❊✖❼❉❼⑩❄❻❆✍❚✡➝✖❿❪ ❆ ❪❊❅➇➌❼✍❹☛❈✚❼ ❈➃❵➃ ❼✍❹❅❚➊❚❪➇❥❚✡❊➈❈➃❿☛❚❴ ❄ ❨ ❼✍❹❅❚➊❺❻❈✚❼✍❆❪✯❽ ➀✩❚ ➁ ❄❴⑩❪❼ ❪✯➈❚ ø ❸✱❄ ❴❹❅❄✚➂➉❼⑩❹❪➅❴ ➂➞❚➊❊❅❄❋❼✍❚➄❼⑩❹☛❈❋❼ ❪❨ ë❫❪❵❴ ❴●✖❺➌❺➌❚❱❼✍❆❪ ❯❝❈❋❊✳❦❂❹✳❈❴ ❆✍❚✡❈➃ ❯❱❄✆❚❱➟❩❯❪❚✡❊❇❼ ❴ ❳ ❪❼❢❯✡❈❋❊✦➀✩❚➊➂➎❆❪❼②❼✍❚✘❊➒❈❴➐↔ ì➉➠▼✱➡➠ ❳ ➂➎❹❅❚✡❆⑩❚ ➡ ❪➅❴ ❼⑩❹☛❚➢❦❪❈❋➇❇❄❥❊☛❈➃ ❺❻❈✚❼⑩❆ ❪✯❽ ❄ ❨ ❚❪➇❥❚✘❊➈❈➃❿❅❚ ❴ ❈❋❊☛❦ ➠➤❪❵❴ ❈❋❊→❄❥❆⑩❼⑩❹☛❄❥❊❅❄❇❆⑩❺❻❈➃ ❼⑩❆➥❈❋❊❴ ❨❄❥❆✍❺❻❈✚❼ ❪❄❥❊ ❳ ❪ ø ❚ ø ➠ ☎ ￾ ì➦➠▼ ø ❸❖❹❅❚✘❊ ❳ ❍ ▼↔❍ ì ❍ ▼ ➠▼➡➠❍ ì➉➧▼ ➡➧ ❲ ö❻❨❄❥❆❉❈❋❊✆●❂❍ ❭❬ ö ✞❄❇❆➐❈❋❊✆● ➧ ì❫➠❍ ❭❬ ö ❴②❪❊☛❯❱❚ ➠➨❪❵❴ ❊❅❄❥❊❅↕ ❴⑩❪❊☛➇❥❿➃❈❋❆ ✠ ❳ ❪❺ ➁ ➃❵❪❚ ❴ ❼⑩❹✳❈✚❼✐❈➃✯➃ ❼⑩❹❅❚❾❚✘❊✖❼⑩❆ ❪❚ ❴❉❪❊ ➡ ❺➢❿❴❼✐➀✳❚ ➇❥❆✍❚✡❈❋❼⑩❚✡❆➩❼⑩❹☛❈❥❊➏➙✡❚✘❆✍❄ ø✟➫➀➈✆❪❄❥❿❴⑩➃● ❳ ❈❥❊✆●➢❺❻❈✚❼✍❆❪✯❽ ➂➎❹❪ ❯➥❹ ❪❵❴❖➭✪➯➳➲➉❪➅❴ ❈➃➅❴❄➊❊☛❄❥❊✪↕ ❴②❪❊❅➇❇❿➃❈❥❆➵❈❋❊✳❦ ❼⑩❹☛❚✘❆✍❚❨❄❥❆✍❚ ❪❊➈❚✡❆②❼ ❪➀➃❚ ❳ ↔☎ ￾ ì➉➠➡ ☎ ￾ ➠▼ ø ❜➓✙✜❀❅✾✩✰✳✷✿❀❅✲➞❜❂✰✩✑✔✙✵✷✼❀❅✷✄✕ ➆♦❚ ❴❈④●➌❼✍❹☛❈✚❼➐❈➜❺❻❈✚❼✍❆❪✯❽➓↔ ì✤■✶➸◗➻➺ ▲ ￾r➼ ◗ ➽ ➺ ➽ ➼ ✂ ❪➅❴ ❦❪❈❥➇❥❄❇❊☛❈➃❵➃●❩❦❅❄❥❺❪❊✳❈❋❊✖❼ ❪❨ ➾