An Algebraic Criterion of Zero Solutions of Some Dynamic Systems
نویسندگان
چکیده
and Applied Analysis 3 Proof. Let α1, α2, . . . , αr be the base ofW and αr 1, . . . , αn ofU. If the constraints ofA onW are lIW and U is a k-complement space of A for W , then Aαi lαi, i 1, . . . , r and there exists a1j , . . . , arj making Aαj − kαj a1jα1 · · · arjαr , that is, Aαj a1jα1 · · · arjαr kαj , j r 1, . . . , n. So A α1, . . . , αr , αr 1, . . . , αn α1, . . . , αr , αr 1, . . . , αn ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ l · · · 0 a1,r 1 · · · a1n .. . . . .. .. .. 0 · · · l ar,r 1 · · · arn 0 k · · · 0 . . . .. k ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ . 2.5 Because a subspace W is a B invariant subspace, we can write Bαi b1iα1 · · · briαr , i 1, . . . , r, Bαi b1jα1 · · · brjαr · · · bnjαn, j r 1, . . . , n. So B ( α1, . . . , αr , αr 1, . . . , αn ) ( α1, . . . , αr , αr 1, . . . , αn ) ⎛ ⎜⎜⎜⎜⎜⎜⎜⎝ b11 · · · b1r b1,r 1 · · · b1n .. . . . .. .. .. br1 · · · brr br,r 1 · · · brn 0 br 1,r 1 · · · br 1,n .. . . . .. bn,r 1 · · · bnn ⎞ ⎟⎟⎟⎟⎟⎟⎟⎠ .
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