Class Note for MATH 410 at UA
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Date Created: 02/06/15
THE UNIVERSITY a OF ARIZONA Math 410 Matrix Analysis Section 82 Eigenvalues and Eigenvectors Section 83 Eigenvector Basis and Diagonalization Paul Dostert March 27 2009 111 A Eigenpairs Let A be an n gtlt n matrix A scalar A is called an eigenvalue of A if there is a nonzero vector 11 3A 0 called an eigenvector such that Av Av We sometimes refer to A711 as an eigenpair Note We will restrict ourselves to ONLY square matrices A when discussing eigenvalues Thm A scalar A is an eigenvalue of A iff A AI is singular The corresponding eigenvectors are the nonzero solutions to A AI 11 0 Prop A scalar A is an eigenvalue of A iff A is a solution to the characteristic equation det A AI 0 Note There are infinitely many eigenvectors corresponding to an eigenvalue If A711 is an eigenpair then so is A7011 for any 0 e R Ex Prove there are infintely many eigenvectors corresponding to any one eigenvalue A Finding Eigenvalues and Eigenvectors We use the following procedure to find the eigenvalues and eigenvectors eigenspaces of an n gtlt n matrix A 1 Compute the characteristic polynomial det A AI of A 2 Find the eigenvalues of A by solving the characteristic equation det A AI 0 for A 3 For each eigenvalue A find the eigenspace VA this is the null space of A AI by solving Av Av 0 4 Find a basis for each eigenspace 2 0 0 Ex Find the eigenvalues and eigenspaces for A 1 2 1 Do the 1 2 0 same for A l ll lN l ll lN DNO A Multiplicity 1 2 1 Ex Find the eigenvalues and eigenspaces for A 2 0 2 Do 1 2 3 5 2 2 the same for A 2 5 2 2 2 5 Note that in the above examples we only have two distinct eigenvalues If we count with multiplicity then each matrix has 3 eigenvalues We define the algebraic multiplicity of an eigenvalue to be its multiplicity as a root of the characteristic equation For each of the previous examples what is the algebraic multiplicity of the eigenvalues Note that for the first example the eigenspace corresponding to A 2 consists of only 1 vector For the second example the eigenspace corresponding to A 3 has 2 vectors in the eigenspace We define the geometric multiplicity of an eigenvalue to be the the dimension of the eigenspace A Complex Eigenvalues amp Eigenvectors Ex Find the eigenvalues and eigenvectors of 2 1 2 A 1 0 1 1 1 0 Prop If A is a real matrix with complex eigenvalue A u w and corresponding complex eigenvector v c iy then A u w and 17 c z y are also an eigenpair Ex Prove this A Characteristic Polynomial The characteristic polynomial of A is pAdetA Ian clco Note the pA 0 det A co so the constant term in the characteristic polynomial is the determinant For a 2 gtlt 2 matrix A lt Z Z the characteristic polynomial is pA a A d A bc A2 adad bc A2 trAAdetA where trA is the trace ofA which is the sum of diagonal elements A well known theorem in algebra states that we can factor a characteristic polynomial in terms of its roots PAOO 1ngt gt1quot39 n Recall There is no explicit way to find the roots of polynomials of degree 5 and higher A Eigenvalues Trace and Determinant Prop The sum of the eigenvalues of a matrix is equal to its trace A139ntl ACL11Cbnn The product of the eigenvalues equals its determinant A1n detA Note Repeated eigenvalues are counted multiple times in each of the above formulas O 1 2 2 0 Ex Verify the above proposition for A 1 2 1 0 A Complete Matrix Lemma If A17 7 Ak are distinct eigenvalues of A then the corresponding eigenvectors v17 711k are linearly independent Thm If the n gtlt n matrix A has n distinct real eigenvalues A17 7 An then the corresponding eigenvectors v17 7 Un form a basis of Rquot If A has n distinct complex eigenvalues then the v7 form a basis of Cquot Note The converse is not necessarily true An eigenvalue A of A is called complete if the corresponding eigenspace VA has the same dimension as its multiplicitiy The matrix A is complete if all its eigenvalues are Another way this is stated is the geometric and algebraic multiplicity of each eigenvalue must be the same Complete matrices are often called perfect matrices 1 0 1 Ex Is the matrix A 1 1 0 complete A Diagonalization A square matrix A is diagonalizable if there exists a nonsingular matrix S and a diagonal matrix A diag A1 An such that SilAS A or A SASil Note that we can write AS SA which taking the kth column of this equation gives us Auk Akvk So S contains the eigenvectors of A as columns and A contains the corresponding eigenvalues In order for S 1 to exists the columns of S must by linearly independent implying the eigenvectors are linearly independent and form a basis for R Thm A matrix is complex diagonalizable iff it is complete A matrix is real diagonalizable iff it is complete and has all real eigenvalues 1 0 1 Ex Determine S and A for A 1 1 0 1 0 1 JZL Matlab Examples Finding Epairs Finding eigenpairs is quite easy in Matlab You simply need to create a matrix then call the eig function It returns two matrices The 15t contains the eigenvectors as columns and the 2 d is a diagonal matrix that contains the corresponding eigenvalues To find the eigenvalues and vectors of A 3 we do A 3 0 8 1 V D eigA d diagD This makes d a vector of the eigenvalues We can verify these are eigenvalues and eigenvectors by computing Ax and Ax AV1 d1V1 AV2 d2V2 A Matlab Eigenspaces amp Multiplicity Finding the algebraic multiplicity of an eigenvalue is easy but how would we find the geometric multiplicity We can do this by reducing A A to RREF 5 2 2 Consider A 2 5 2 We write 2 2 5 A52 225 2 2 25 v D eigA rrefA eye33D11 rrefA eye33D33 We find that the geometric multiplicity for A 9 is 1 since the rref has one zero row and the geometric multiplicity for A 3 is 2 since the rref has two zero rows
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