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Eigenvalues and Eigenvectors, Algebraic Multiplicity is The number of times a particular eigenvalue appears in the solution set of the characteristic polynomial, Eigenvectors are special vectors associated with a matrix and its eigenvalues which satisfy A x = λ x, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> A= </mtext> <mfenced open="[" close="]"> <mtext> 3 1 1 3 </mtext> </mfenced> <mtext> , then </mtext> <mfenced open="[" close="]"> <mtext> 1 1 </mtext> </mfenced> <mtext> and </mtext> <mfenced open="[" close="]"> <mtext> 1 -1 </mtext> </mfenced> <mtext> are eigenvectors with eigenvalues 4 and 2 </mtext> </mrow> </math> since <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mfenced open="[" close="]"> <mtext> 3 1 1 3 </mtext> </mfenced> <mfenced open="[" close="]"> <mtext> 1 1 </mtext> </mfenced> <mtext> =4 </mtext> <mfenced open="[" close="]"> <mtext> 1 1 </mtext> </mfenced> </mrow> </math>, Eigenvalues can be used to compute Determinants, A x = λ x e.g. <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> A= </mtext> <mfenced open="[" close="]"> <mtext> 3 1 1 3 </mtext> </mfenced> <mtext> , then </mtext> <mfenced open="[" close="]"> <mtext> 1 1 </mtext> </mfenced> <mtext> and </mtext> <mfenced open="[" close="]"> <mtext> 1 -1 </mtext> </mfenced> <mtext> are eigenvectors with eigenvalues 4 and 2 </mtext> </mrow> </math>, Eigenvectors of related matrices possess interesting properties, such as <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> Given Av=λv, then </mtext> <mmultiscripts> <mtext> A </mtext> <none/> <mtext> n </mtext> </mmultiscripts> <mtext> v= </mtext> <mmultiscripts> <mtext> λ </mtext> <none/> <mtext> n </mtext> </mmultiscripts> <mtext> v for any integer value of n </mtext> </mrow> </math>, Eigenvalues possess Important Features of Their Matrix, p(λ)=det(A-λI)=0 e.g, <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> A= </mtext> <mfenced open="[" close="]"> <mtext> 3 1 1 3 </mtext> </mfenced> <mtext> , p(λ)= </mtext> <mmultiscripts> <mtext> λ </mtext> <none/> <mtext> 2 </mtext> </mmultiscripts> <mtext> -6λ+8=0 ⇒λ=4 or 2 are eigenvalues of A </mtext> </mrow> </math>, Eigenvectors of related matrices possess interesting properties, such as <math xmlns="http://www.w3.org/1998/Math/MathML"> <mrow> <mtext> If A is invertible (i.e. λ≠0), then its inverse matrix has the same eigenvectors with different associated eigenvalues 1/λ. </mtext> </mrow> </math>, Eigenvalues are special numbers which satisfy A x = λ x
