Diagonalization Similar Matrices We have seen that the commutative property does not hold for matrices, so that if A is an n x n matrix, then          P-1AP is not necessarily equal to A.  For different nonsingular matrices P, the above expression will represent different matrices.  However, all such matrices share some important properties as we shall soon see. Definition Let A and B be an n x n matrices, then A is similar to B if there is a nonsingular matrix P with         B  =  P-1AP   Example Consider the matrices         Then          is similar to A. Notice the three following facts A is similar to A. If A is similar to B then B is similar to A. If A is similar to B and B is similar to C then A is similar to C.   We call a relationship with these three properties an equivalence relationship.  We will prove the third property. If A is similar to B and B is similar to C then there are matrices P and Q with         B  =  P-1AP        and        C  =  Q-1BQ We need to find a matrix R with          C  =  R-1AR We have         C  =  Q-1BQ  =  Q-1(P-1AP)Q  =           (Q-1P-1)A(PQ)  =  (PQ)-1A(PQ)  =  R-1AR There is a wonderful fact that we state below.   Theorem If A and B are similar matrices, then they have the same eigenvalues.   Proof It is enough to show that they have the same characteristic polynomials.  We have         det(lI - B)  =  det(lI - P-1AP)  =  det(P-1lIP - P-1AP)         =  det(P-1(lI - A)P)  =  det(lI - A) Diagonalized Matrices The easist kind of matrices to deal with are diagonal matrices.  Determinants are simple, the eigenvalues are just the diagonal entries and the eigenvectors are just elements of the standard basis.  Even the inverse is a piece of cake (if the matrix is nonsingular).  Although most matrices are not diagonal, many are diagonalizable, that is they are similar to a diagonal matrix.  Definition A matrix A is diagonalizable if A is similar to a diagonal matrix D.         D  =  P-1AP  The following theorem tells us when a matrix is diagonalizable and if it is how to find its similar diagonal matrix D. Theorem Let A be an n x n matrix.  Then A is diagonalizable if and only if A has n linearly independent eigenvectors.  If so, then          D  =  P-1AP If {v1, ... , vn} are the eigenvectors of A and {l1, ... , ln} are the corresponding eigenvalues, then          vj the jth column of P  and          [D]jj  =  lj    Example In the last discussion, we saw that the matrix         has -1 and 4 as eigenvalues with associated eigenvectors         Hence         You can verify that         D  =  P-1AP Proof of the Theorem If          D  =  P-1AP for some diagonal matrix D and nonsingular matrix P, then         AP  =  PD Let vi be the jth column of P and [D]jj  =  lj.  Then the jth column of AP is Avi and the jth column of PD is livj.  Hence         Avj  =  livj  so that vj is an eigenvector of A with corresponding eigenvalue lj.  Since P has its columns as eigenvectors, and P is nonsingular, rank(P)  =  n, and the columns of P (the eigenvalues of A) are linearly independent. Next suppose that the eigenvalues of A are linearly independent.  Then form D and P as above.  Then since          Avj  =  livj  The  jth column of AP equals the jth column of PD, hence AP  =  PD.  Since the columns of P are linearly independent, P is nonsingular so that         D  =  P-1AP Theorem Let A be an n x n matrix with n real and distinct eigenvalues.  Then A is diagonalizable.   Proof Let          {l1, ... , lk}   and    {v1, ... , vk} with          rank(Span({v1, ... , vk}))  =  k - 1   be the eigenvalues and eigenvectors of A.  We need to show that none of the vectors can be written as a linear combination of the rest.  Without loss of generality, we need show that the first can not be written as a linear combination of the rest.  If          v1 = c2v2 + ... + cnvk         (1) We can multiply both sides of the equation by A to get           l1v1 = Ac2v2 + ... + Acnvk  =  c2l2v2 + ... + cnlnvk         (2) Multiply (1) by l1 and subtract it from (2) to get         c2(l2 - l1)v2 + ... + cn(ln - l1)vn = 0 Since the l's are distinct, the ci's must all be zero, which is a contradiction (otherwise the rank would be less than k - 1).  Hence           rankSpan({v1, ... , vk})  =  k for any k.  In particular, let k  =  n, and the result follows.   Note that the converse certainly does not hold.  For example, the identity matrix I has 1 as all of its eigenvalues, but it is diagonalizable (it is diagonal).   Steps to Diagonalize a Matrix Find the eigenvalues by finding the roots of the characteristic polynomial. Find the eigenvectors by finding the null space of A - liI. If the number of linearly independent vectors is n, then let P be the matrix whose columns are eigenvectors and let D be the diagonal matrix with [D]jj  =  lj Example Diagonalize the matrix         Solution We find the characteristic polynomial         The roots are 1 (with multiplicity 2) and 2 (with multiplicity 1). Now we find the eigenspaces associated with the eigenvalues.  We have         A basis for the null space is          Next we find a basis for the eigenspace associated with the eigenvalue 2.  We have         A basis for this null space is         Now put this all together to get           Back to the Linear Algebra Home Page