Linear-2 2

Let A∈Rn×nProve or disprove: A invertible⟺AAT invertibleProof:det(A)≠0⟺det(AT)≠0⟺det(AAT)≠0
Let A∈Rn×n be symmetricProve or disprove: A invertible⟺A+AT invertibleProof:A+AT=2Adet(A)≠0⟺det(2A)=2ndet(A)≠0⟺det(A+AT)≠0
Let α∈RLet A∈Rn×nAij={αi=j1i≠j|α1…11α…1⋮⋮⋱⋮1…1α|→∀i∈[2,n]:R1+Ri|α+n−1α+n−1…α+n−11α…1⋮⋮⋱⋮1…1α|→1α+n−1R1(α+n−1)|11…11α…1⋮⋮⋱⋮1…1α|→∀i∈[2,n]:Ri−R1(α+n−1)|11…10α−1…0⋮⋱⋱⋮0…0α−1|⟹det(A)=(α+n−1)(α−1)n−1
Let A∈Fn×nLet λ∈Fλ is called an eigenvalue of A⟺∃v∈Fn≠0:Av=λAv is then called an eigenvector of A in respect to eigenvalue λλ≠0⟹A(1λv)=v⟹v∈C(A)λ=0⟹Av=0⟹v∈N(A)

Characteristic polynomial

Characteristic polynomial PA(λ)=|λI−A||λ−56−3λ+4|=(λ−5)(λ+4)+18=(λ+1)(λ−2)PA(λ)=0⟺λ is an eigenvalue of AProof:∃v≠0:Av=λv⟺(λI−A)v=0⟺v∈N(λI−A)⟺det(λI−A)=0⟺PA(λ)=0
A=(01−10)PA(λ)=|λ−11λ|=λ2+1A∈Rn×n⟹No eigenvaluesA∈Cn×n⟹±i is an eigenvalueA∈Z2n×n⟹1 is an eigenvalue
A=(311242113)|λ−3−1−1−2λ−4−2−1−1λ−3|→R1+R2+R3|λ−6λ−6λ−6−2λ−4−2−1−1λ−3|=(λ−6)|111−2λ−4−2−1−1λ−3|→(λ−6)|1110λ−2−200λ−2|=(λ−6)(λ−2)2PA(λ)=0⟺[λ=6λ=2λ=6⟹vλ=N(000−22−2−1−13)=N(000−11−10−12)=N(000−1010−12)==sp{(121)}λ=2⟹vλ=N(−1−1−1−2−2−2−1−1−1)=N(111000000)=sp{(−110),(−101)}
Let A∼BPA(λ)=PB(λ)Proof:|λI−B|=|λI−P−1AP|=|P−1(λI−A)P|==|P−1|⋅|λI−A|⋅|P|=|λI−A|
Let T:V→V be a linear transformation∀B basis of V:λ is an eigenvalue of T⟺λ is an eigenvalue of [T]BProof:Let v≠0∈VTv=λv⟺[Tv]B=[λv]B⟺[T]B[v]B=λ[v]B