Linear-2 2

Commutativity of determinant #lemma

∀A,B∈Fn×n|AB|=|BA|Proof:|AB|=|A|⋅|B|=|B|⋅|A|=|BA|

Determinant of n matrix product #theorem

∀{Ai}i∈I⊆Fn×n|∏i∈IAi|=∏i∈I|Ai|Proof:By induction, starting from the identical lemma for two matrices

Determinant of transpose #lemma

∀A∈Fn×n|A|=|AT|Proof:Let P:αRiP(I) is diagonal⟹P(I)=P(I)T⟹|P(I)|=|P(I)T|Let P:Ri↔RjP(I)=P(I)T⟹|P(I)|=|P(I)T|Let P:Ri+αRjP(I) is triangular⟹P(I)T is triangularDiagonals of P(I) and P(I)T are the same⟹|P(I)|=|P(I)T|Let A be non-invertible⟹rank(AT)=rank(A)<n⟹AT is non-invertible⟹|AT|=0=|A|Let A be invertible⟹A=∏i=1kPi(I)⟹|A|=|∏i=1kPi(I)|=∏i=1k|Pi(I)|AT=(∏i=1kPi(I))T=∏i=1kPk+1−i(I)T⟹|AT|=|∏i=1kPk+1−i(I)T|=∏i=1k|Pk+1−i(I)T|==∏i=1k|Pk+1−i(I)|=∏i=1k|Pi(I)|⟹|AT|=∏i=1k|Pi(I)|=|A|

Corollary

Elementary column operations affect determinant in the same way row operations do

Determinant of an inverse #lemma

Let A∈Fn×nLet ∃A−1⟹|A|≠0|AA−1|=|A|⋅|A−1|=|I|=1⟹|A−1|=1|A|

Summary

∃A−1⟺|A|≠0|AB|=|A|⋅|B||A|=|AT||αA|=αn|A||A−1|=|A|−1

Matrix minor #definition

Let A∈Fn×nLet i,j∈[1,n]Minor Mij(A) is a matrix,obtained by removing row i and column j from matrix AA=(123456789)⟹M13(A)=(4578)

Using minors to calculate determinant #lemma

Let A∈Fn×n∀i∈[1,n]:|A|=∑j=1n(−1)i+jaij|Mij(A)|∀j∈[1,n]:|A|=∑i=1n(−1)i+jaij|Mij(A)|

Example

A=(123456789)|A|=1|5689|−2|4679|+3|4578|=−3+12−9=0|A|=−2|4679|+5|1379|−8|1346|=12−60+48=0

Determinant of a linear operator (transformation) #lemma

Let V be a vector space over FLet T:V→V be a linear transformationLet B be a basis of V|T|:=|[T]BB|∀B,C basis of V:|[T]BB|=|[T]CC|Proof:[T]BB=[I]BC[T]CC[I]CB⟹|[T]BB|=|[I]BC[T]CC[I]CB|=|[I]BC|⋅|[T]CC|⋅|[I]CB|==|[T]CC|⋅|[I]BC[I]CB|=|[T]CC|⋅|[I]BB|=|[T]CC|

Eigenvalues and eigenvectors #definition

Let A∈Fn×nλ∈F is called an Eigenvalue of A if∃v≠0∈Fn:Av=λvv is then called an Eigenvector of A in respect to Eigenvalue λ

Example

(2304)(32)=(128)=4(32)λ=4,v=(32)

Determinant in relation to eigenvalue #lemma

Let A∈Fn×nλ is an Eigenvalue of A⟺|λI−A|=0Proof:λ is an Eigenvalue of A⟺∃v≠0∈Fn:Av=λv⟺∃v≠0∈Fn:λv−Av=0⟺∃v≠0∈Fn:(λI−A)v=0⟺N(λI−A)≠{0}⟺∄(λI−A)−1⟺|λI−A|=0

Corollary

∄A−1⟺0 is an Eigenvalue of A∃v≠0∈Fn:Av=0v=0⟺N(A)≠{0}⟺∄A−1|A|=0⟺|−A|=0⟺|0I−A|=0⟺0 is an Eigenvalue of A

Example

A=(2304)|λI−A|=|λ−230λ−4|=(λ−2)(λ−4)=0⟹[λ=2λ=4How do we find Eigenvectors in respect to these Eigenvalues?N(λI−A)={v|(λI−A)v=0}⟹Eigenvectors in respect to Eigenvalue λis a set of non-zero solutions to homogeneous system of equations (λI−A)v=0Let λ=2(0−300−20)→(010000)⟹v∈sp{(10)}Let λ=4(2−300)⟹v∈sp{(32)}

Eigenspace #definition

Let A∈Fn×nLet α∈F be an Eigenvalue of ASet of Eigenvectors in respect to α, and zero vector, is then called EigenspaceE={v|(λI−A)v=0}=N(λI−A)

Characteristic polynomial #definition

Let A∈Fn×nLet λ∈FPA(λ)=|λI−A| is called a characteristic polynomial of A

Matrix similarity #definition

Let A,B∈Fn×nMatrices A,B are called similar if∃P∈Fn×n:P−1AP=BSimilarity is an equivalence relation:Reflexive: A=I−1AISymmetric: P−1AP=B⟹B=(P−1)−1AP−1Transitive: B=P−1AP,C=P1−1BP1⟹C=P1−1P−1APP1=(PP1)−1A(PP1)

Similar matrix properties #lemma

Let A,B∈Fn×nLet A∼B|A|=|B|tr(A)=tr(B)rank(A)=rank(B)PA(λ)=PB(λ)

Diagonalizable matrix #definition

Let A∈Fn×nLet D be a diagonal matrix ∈Fn×nA is called diagonalizable⟺∃P∈Fn×n:P−1AP=D