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1、Eigenvalue and Eigenvector5thweek/Linear AlgebraObjectives of This Week2The goal is to understandEigenvectors and eigenvaluesNull spaceCharacteristic equationFinding all eigenvalues and eigenvectors3Eigenvectors and Eigenvalues Definition:An eigenvector of a square matrix is a nonzero vector such th
2、at =for some scalar In this case,is called an eigenvalue of,andsuch an is called an eigenvector corresponding to.4Transformation Perspective Consider a linear transformation x x=x x.If x x is an eigenvector,then x x=x x=,which means the output vector has the same direction as x x,but the length is s
3、caled by a factor of.Example:For =2653,an eigenvector is 11since x x=x x=265311=88=811x x=8 8=8115Computational Advantage Which computation is faster between 265311and 811?6Eigenvectors and Eigenvalues The equation =can be re-written as =is an eigenvalue of an matrix if and only if this equation has
4、 a nontrivial solution(since should be a nonzero vector).7Eigenvectors and Eigenvalues =The set of all solutions of the above equation is the null space of the matrix ,which we call the eigenspace of corresponding to.The eigenspace consists of the zero vector and all the eigenvectors corresponding t
5、o,satisfying the above equation.8Null Space Definition:The null space of a matrix is the set of all solutions of =called a homogeneous linear system.We denote the null space of as Nul.For =12,should satisfy the following:1=0,2=0,=0 That is,should be orthogonal to every row vector in.9Null Space is a
6、 Subspace Theorem:The null space of a matrix is a subspace of.In other words,the set of all the solutions of a system =is a subspace of.Note:An eigenspace thus have a set of basis vectorswith a particular dimension.10Example:Eigenvalues and Eigenvectors Example:Show that 8 is an eigenvalue of a matr
7、ix =2653and find the corresponding eigenvectors.Solution:The scalar 8 is an eigenvalue of if and only if the equation 8 =has a nontrivial solution:The solution is =11for any nonzero scalar,which is Span11.8 =6655=11Example:Eigenvalues and Eigenvectors In the previous example,3 is also an eigenvalue:
8、+3 =5656=The solution is =15/6for any nonzero scalar,which is Span15/6.12Characteristic Equation How can we find the eigenvalues such as 8 and 3?If =has a nontrivial solution,then the columns of should be noninvertible.If it is invertible,cannot be a nonzero vector since 1 =1 =Thus,we can obtain eig
9、envalues by solving det =0called a characteristic equation.Also,the solution is not unique,and thus has linearly dependent columns.13Example:Characteristic Equation In the previous example,=2653is originally invertible since det =det2653=6 30=24 0.By solving the characteristic equation,we want to fi
10、nd that makes non-invertible:det =det2 653 =2 3 30=2 5 25=8 3 =0=3 or 814Example:Characteristic Equation Once obtaining eigenvalues,we compute the eigenvectors for each by solving =15Eigenspace Note that the dimension of the eigenspace(corresponding to a particular)can be more than one.In this case,
11、any vector in the eigenspace satisfies x x=x x=33Multiplication by acts as a dilation on the eigenspace16Finding all eigenvalues and eigenvectors In summary,we can find all the possible eigenvalues and eigenvectors,as follows.First,find all the eigenvalue by solving the characteristic equation:det =0 Second,for each eigenvalue,solve for =and obtain the set of basis vectors of the corresponding eigenspace.Eigenvectors and eigenvaluesCharacteristic equationFinding all eigenvalues and eigenvectorsSummary