Section SD  Similarity and Diagonalization

From A First Course in Linear Algebra
Version 1.08
© 2004.
Licensed under the GNU Free Documentation License.
http://linear.ups.edu/

This section’s topic will perhaps seem out of place at first, but we will make the connection soon with eigenvalues and eigenvectors. This is also our first look at one of the central ideas of Chapter R.

Subsection SM: Similar Matrices

The notion of matrices being “similar” is a lot like saying two matrices are row-equivalent. Two similar matrices are not equal, but they share many important properties. This section, and later sections in Chapter R will be devoted in part to discovering just what these common properties are.

First, the main definition for this section.

Definition SIM
Similar Matrices
Suppose A and B are two square matrices of size n. Then A and B are similar if there exists a nonsingular matrix of size n, S, such that A = S1BS.

We will say “A is similar to B via S” when we want to emphasize the role of S in the relationship between A and B. Also, it doesn’t matter if we say A is similar to B, or B is similar to A. If one statement is true then so is the other, as can be seen by using S1 in place of S (see Theorem SER for the careful proof). Finally, we will refer to S1BS as a similarity transformation when we want to emphasize the way S changes B. OK, enough about language, let’s build a few examples.

Example SMS5
Similar matrices of size 5
If you wondered if there are examples of similar matrices, then it won’t be hard to convince you they exist. Define

B = 4132 2 1 21 3 2 41 3 2 2 34213 3 11 1 4 S = 1 2 1 1 1 0 1 121 1 3 1 1 1 23 3 1 2 1 3 1 2 1

Check that S is nonsingular and then compute

A = S1BS = 10 1 0 2 5 1 0 1 0 0 3 0 2 1 3 0 0 1 0 1 41 1 1 1 4132 2 1 21 3 2 41 3 2 2 34213 3 11 1 4 1 2 1 1 1 0 1 121 1 3 1 1 1 23 3 1 2 1 3 1 2 1 = 1027298025 2 6 6 10 2 3 11 9 14 9 1 13 0 10 1 11 35 6 49 19

So by this construction, we know that A and B are similar.

Let’s do that again.

Example SMS3
Similar matrices of size 3
Define

B = 1384 12 7 4 24 16 7 S = 1 1 2 213 1 2 0

Check that S is nonsingular and then compute

A = S1BS = 641 321 5 3 1 1384 12 7 4 24 16 7 1 1 2 213 1 2 0 = 10 0 0 3 0 0 01

So by this construction, we know that A and B are similar. But before we move on, look at how pleasing the form of A is. Not convinced? Then consider that several computations related to A are especially easy. For example, in the spirit of Example DUTM, det A = (1)(3)(1) = 3. Similarly, the characteristic polynomial is straightforward to compute by hand, pA x = (1 x)(3 x)(1 x) = (x 3)(x + 1)2 and since the result is already factored, the eigenvalues are transparently λ = 3, 1. Finally, the eigenvectors of A are just the standard unit vectors (Definition SUV).

Subsection PSM: Properties of Similar Matrices

Similar matrices share many properties and it is these theorems that justify the choice of the word “similar.” First we will show that similarity is an equivalence relation. Equivalence relations are important in the study of various algebras and can always be regarded as a kind of weak version of equality. Sort of alike, but not quite equal. The notion of two matrices being row-equivalent is an example of an equivalence relation we have been working with since the beginning of the course (see Exercise RREF.T11). Row-equivalent matrices are not equal, but they are a lot alike. For example, row-equivalent matrices have the same rank. Formally, an equivalence relation requires three conditions hold: reflexive, symmetric and transitive. We will illustrate these as we prove that similarity is an equivalence relation.

Theorem SER
Similarity is an Equivalence Relation
Suppose A, B and C are square matrices of size n. Then

  1. A is similar to A. (Reflexive)
  2. If A is similar to B, then B is similar to A. (Symmetric)
  3. If A is similar to B and B is similar to C, then A is similar to C. (Transitive)

Proof   To see that A is similar to A, we need only demonstrate a nonsingular matrix that effects a similarity transformation of A to A. In is nonsingular (since it row-reduces to the identity matrix, Theorem NMRRI), and

In1AI n = InAIn = A

If we assume that A is similar to B, then we know there is a nonsingular matrix S so that A = S1BS by Definition SIM. By Theorem MIMI, S1 is invertible, and by Theorem NI is therefore nonsingular. So

(S1)1A(S1) = SAS1  Theorem MIMI = SS1BSS1  Definition SIM = SS1 B SS1  Theorem MMA = InBIn  Definition MI = B  Theorem MMIM

and we see that B is similar to A.

Assume that A is similar to B, and B is similar to C. This gives us the existence of two nonsingular matrices, S and R, such that A = S1BS and B = R1CR, by Definition SIM. (Notice how we have to assume SR, as will usually be the case.) Since S and R are invertible, so too RS is invertible by Theorem SS and then nonsingular by Theorem NI. Now

(RS)1C(RS) = S1R1CRS  Theorem SS = S1 R1CRS  Theorem MMA = S1BS  Definition SIM = A

so A is similar to C via the nonsingular matrix RS.

Here’s another theorem that tells us exactly what sorts of properties similar matrices share.

Theorem SMEE
Similar Matrices have Equal Eigenvalues
Suppose A and B are similar matrices. Then the characteristic polynomials of A and B are equal, that is pA x = pB x.

Proof   Suppose A and B have size n and are similar via the nonsingular matrix S, so A = S1BS by Definition SIM.

pA x = det A xIn  Definition CP = det S1BS xI n  Definition SIM = det S1BS xS1I nS  Theorem MMIM = det S1BS S1xI nS  Theorem MMSMM = det S1 B xI n S  Theorem MMDAA = det S1 det B xI n det S  Theorem DRMM = det S1 det S det B xI n  Property MCCN = det S1S det B xI n  Theorem DRMM = det In det B xIn  Definition MI = 1 det B xIn  Definition DM = pB x  Definition CP

So similar matrices not only have the same set of eigenvalues, the algebraic multiplicities of these eigenvalues will also be the same. However, be careful with this theorem. It is tempting to think the converse is true, and argue that if two matrices have the same eigenvalues, then they are similar. Not so, as the following example illustrates.

Example EENS
Equal eigenvalues, not similar
Define

A = 11 01 B = 10 01

and check that

pA x = pB x = 1 2x + x2 = (x 1)2

and so A and B have equal characteristic polynomials. If the converse of Theorem SMEE were true, then A and B would be similar. Suppose this is the case. In other words, there is a nonsingular matrix S so that A = S1BS. Then

A = S1BS = S1I 2S = S1S = I 2A

and this contradiction tells us that the converse of Theorem SMEE is false.

Subsection D: Diagonalization

Good things happen when a matrix is similar to a diagonal matrix. For example, the eigenvalues of the matrix are the entries on the diagonal of the diagonal matrix. And it can be a much simpler matter to compute high powers of the matrix. Diagonalizable matrices are also of interest in more abstract settings. Here are the relevant definitions, then our main theorem for this section.

Definition DIM
Diagonal Matrix
Suppose that A is a square matrix. Then A is a diagonal matrix if Aij = 0 whenever ij.

Definition DZM
Diagonalizable Matrix
Suppose A is a square matrix. Then A is diagonalizable if A is similar to a diagonal matrix.

Example DAB
Diagonalization of Archetype B
Archetype B has a 3 × 3 coefficient matrix

B = 7612 5 5 7 1 0 4

and is similar to a diagonal matrix, as can be seen by the following computation with the nonsingular matrix S,

S1BS = 532 3 2 1 1 1 1 1 7612 5 5 7 1 0 4 532 3 2 1 1 1 1 = 111 2 3 1 12 1 7612 5 5 7 1 0 4 532 3 2 1 1 1 1 = 100 0 10 0 02

Example SMS3 provides yet another example of a matrix that is subjected to a similarity transformation and the result is a diagonal matrix. Alright, just how would we find the magic matrix S that can be used in a similarity transformation to produce a diagonal matrix? Before you read the statement of the next theorem, you might study the eigenvalues and eigenvectors of Archetype B and compute the eigenvalues and eigenvectors of the matrix in Example SMS3.

Theorem DC
Diagonalization Characterization
Suppose A is a square matrix of size n. Then A is diagonalizable if and only if there exists a linearly independent set S that contains n eigenvectors of A.

Proof   ( ) Let S = x1,x2,x3,,xn be a linearly independent set of eigenvectors of A for the eigenvalues λ1,λ2,λ3,,λn. Recall Definition SUV and define

R = x1|x2|x3||xn D = λ1 0 0 0 0 λ2 0 0 0 0 λ3 0 0 0 0 λn = [λ1e1|λ2e2|λ3e3||λnen]

The columns of R are the vectors of the linearly independent set S and so by Theorem NMLIC the matrix R is nonsingular. By Theorem NI we know R1 exists.

R1AR = R1A x 1|x2|x3||xn = R1[Ax 1|Ax2|Ax3||Axn]  Definition MM = R1[λ 1x1|λ2x2|λ3x3||λnxn]  xi eigenvector of A for λi = R1[λ 1Re1|λ2Re2|λ3Re3||λnRen]  Definition MVP = R1[R(λ 1e1)|R(λ2e2)|R(λ3e3)||R(λnen)] Theorem MMSMM = R1R[λ 1e1|λ2e2|λ3e3||λnen]  Definition MM = InD  Definition MI = D  Theorem MMIM

This says that A is similar to the diagonal matrix D via the nonsingular matrix R. Thus A is diagonalizable (Definition DZM).

( ) Suppose that A is diagonalizable, so there is a nonsingular matrix of size n

T = y1|y2|y3||yn  and a diagonal matrix (recall Definition SUV) E = d1 0 0 0 0 d2 0 0 0 0 d3 0 0 0 0 dn = [d1e1|d2e2|d3e3||dnen]  

such that T1AT = E. Then consider,

[Ay1|Ay2|Ay3||Ayn] = A y1|y2|y3||yn  Definition MM = AT = InAT  Theorem MMIM = TT1AT  Definition MI = TE  Substitution = T[d1e1|d2e2|d3e3||dnen] = [T(d1e1)|T(d2e2)|T(d3e3)||T(dnen)] Definition MM = [d1Te1|d2Te2|d3Te3||dnTen]  Definition MM = [d1y1|d2y2|d3y3||dnyn]  Definition MVP This equality of matrices (Definition ME) allows us to conclude that the individual columns are equal vectors (Definition CVE). That is, Ayi = diyi for 1 i n. In other words, yi is an eigenvector of A for the eigenvalue di, 1 i n. (Why can’t yi = 0?). Because T is nonsingular, the set containing T’s columns, S = y1,y2,y3,,yn, is a linearly independent set (Theorem NMLIC). So the set S has all the required properties.

Notice that the proof of Theorem DC is constructive. To diagonalize a matrix, we need only locate n linearly independent eigenvectors. Then we can construct a nonsingular matrix using the eigenvectors as columns (R) so that R1AR is a diagonal matrix (D). The entries on the diagonal of D will be the eigenvalues of the eigenvectors used to create R, in the same order as the eigenvectors appear in R. We illustrate this by diagonalizing some matrices.

Example DMS3
Diagonalizing a matrix of size 3
Consider the matrix

F = 1384 12 7 4 24 16 7

of Example CPMS3, Example EMS3 and Example ESMS3. F’s eigenvalues and eigenspaces are

λ = 3 F 3 = 1 2 1 2 1 λ = 1 F 1 = 2 3 1 0 , 1 3 0 1

Define the matrix S to be the 3 × 3 matrix whose columns are the three basis vectors in the eigenspaces for F,

S = 1 22 31 3 1 2 1 0 1 0 1

Check that S is nonsingular (row-reduces to the identity matrix, Theorem NMRRI or has a nonzero determinant, Theorem SMZD). Then the three columns of S are a linearly independent set (Theorem NMLIC). By Theorem DC we now know that F is diagonalizable. Furthermore, the construction in the proof of Theorem DC tells us that if we apply the matrix S to F in a similarity transformation, the result will be a diagonal matrix with the eigenvalues of F on the diagonal. The eigenvalues appear on the diagonal of the matrix in the same order as the eigenvectors appear in S. So,

S1FS = 1 22 31 3 1 2 1 0 1 0 1 1 1384 12 7 4 24 16 7 1 22 31 3 1 2 1 0 1 0 1 = 6 4 2 311 641 1384 12 7 4 24 16 7 1 22 31 3 1 2 1 0 1 0 1 = 3 0 0 01 0 0 0 1

Note that the above computations can be viewed two ways. The proof of Theorem DC tells us that the four matrices (F, S, F1 and the diagonal matrix) will interact the way we have written the equation. Or as an example, we can actually perform the computations to verify what the theorem predicts.

The dimension of an eigenspace can be no larger than the algebraic multiplicity of the eigenvalue by Theorem ME. When every eigenvalue’s eigenspace is this large, then we can diagonalize the matrix, and only then. Three examples we have seen so far in this section, Example SMS5, Example DAB and Example DMS3, illustrate the diagonalization of a matrix, with varying degrees of detail about just how the diagonalization is achieved. However, in each case, you can verify that the geometric and algebraic multiplicities are equal for every eigenvalue. This is the substance of the next theorem.

Theorem DMFE
Diagonalizable Matrices have Full Eigenspaces
Suppose A is a square matrix. Then A is diagonalizable if and only if γA λ = αA λ for every eigenvalue λ of A.

Proof   Suppose A has size n and k distinct eigenvalues, λ1,λ2,λ3,,λk.

( ) Let Si = xi1,xi2,xi3,,xiγAλi, be a basis for the eigenspace of λi, A λi, 1 i k. Then

S = S1 S2 S3 Sk

is a set of eigenvectors for A. A vector cannot be an eigenvector for two different eigenvalues (see Exercise EE.T20) so the sets Si have no vectors in common. Thus the size of S is

i=1kγ A λi = i=1kα A λi  Hypothesis = n  Theorem NEM

We now want to show that S is a linearly independent set. So we will begin with a relation of linear dependence on S, using doubly-subscripted scalars and eigenvectors,

0 = a11x11 + a12x12 + + a1γAλ1x1γAλ1 + a21x21 + a22x22 + + a2γAλ2x2γAλ2 + + ak1xk1 + ak2xk2 + + akγAλkxkγAλk

Define the vectors yi, 1 i k by

y1 = a11x11 + a12x12 + a13x13 + + aγA1λ1x1γAλ1 y2 = a21x21 + a22x22 + a23x23 + + aγA2λ2x2γAλ2 y3 = a31x31 + a32x32 + a33x33 + + aγA3λ3x3γAλ3 yk = ak1xk1 + ak2xk2 + ak3xk3 + + aγAkλkxkγAλk

Then the relation of linear dependence becomes

0 = y1 + y2 + y3 + + yk

Since the eigenspace A λi is closed under vector addition and scalar multiplication, yi A λi, 1 i k. Thus, for each i, the vector yi is an eigenvector of A for λi, or is the zero vector. Recall that sets of eigenvectors whose eigenvalues are distinct form a linearly independent set by Theorem EDELI. Should any (or some) yi be nonzero, the previous equation would provide a nontrivial relation of linear dependence on a set of eigenvectors with distinct eigenvalues, contradicting Theorem EDELI. Thus yi = 0, 1 i k.

Each of the k equations, yi = 0 is a relation of linear dependence on the corresponding set Si, a set of basis vectors for the eigenspace A λi, which is therefore linearly independent. From these relations of linear dependence on linearly independent sets we conclude that the scalars are all zero, more precisely, aij = 0, 1 j γA λi for 1 i k. This establishes that our original relation of linear dependence on S has only the trivial relation of linear dependence, and hence S is a linearly independent set.

We have determined that S is a set of n linearly independent eigenvectors for A, and so by Theorem DC is diagonalizable.

( ) Now we assume that A is diagonalizable. Aiming for a contradiction (Technique CD), suppose that there is at least one eigenvalue, say λt, such that γA λt αA λt. By Theorem ME we must have γA λt < αA λt, and γA λi αA λi for 1 i k, it.

Since A is diagonalizable, Theorem DC guarantees a set of n linearly independent vectors, all of which are eigenvectors of A. Let ni denote the number of eigenvectors in S that are eigenvectors for λi, and recall that a vector cannot be an eigenvector for two different eigenvalues (Exercise EE.T20). S is a linearly independent set, so the the subset Si containing the ni eigenvectors for λi must also be linearly independent. Because the eigenspace A λi has dimension γA λi and Si is a linearly independent subset in A λi, ni γA λi, 1 i k. Now,

n = n1 + n2 + n3 + + nt + + nk  Size of S γA λ1 + γA λ2 + γA λ3 + + γA λt + + γA λk  Si linearly independent < αA λ1 + αA λ2 + αA λ3 + + αA λt + + αA λk Assumption about λt = n  Theorem NEM

This is a contradiction (we can’t have n < n!) and so our assumption that some eigenspace had less than full dimension was false.

Example SEE, Example CAEHW, Example ESMS3, Example ESMS4, Example DEMS5, Archetype B, Archetype F, Archetype K and Archetype L are all examples of matrices that are diagonalizable and that illustrate Theorem DMFE. While we have provided many examples of matrices that are diagonalizable, especially among the archetypes, there are many matrices that are not diagonalizable. Here’s one now.

Example NDMS4
A non-diagonalizable matrix of size 4
In Example EMMS4 the matrix

B = 2 1 24 12 1 4 9 6 5 24 3 4 5 10

was determined to have characteristic polynomial

pB x = (x 1)(x 2)3

and an eigenspace for λ = 2 of

B 2 = 1 2 1 1 2 1

So the geometric multiplicity of λ = 2 is γB 2 = 1, while the algebraic multiplicity is αB 2 = 3. By Theorem DMFE, the matrix B is not diagonalizable.

Archetype A is the lone archetype with a square matrix that is not diagonalizable, as the algebraic and geometric multiplicities of the eigenvalue λ = 0 differ. Example HMEM5 is another example of a matrix that cannot be diagonalized due to the difference between the geometric and algebraic multiplicities of λ = 2, as is Example CEMS6 which has two complex eigenvalues, each with differing multiplicities. Likewise, Example EMMS4 has an eigenvalue with different algebraic and geometric multiplicities and so cannot be diagonalized.

Theorem DED
Distinct Eigenvalues implies Diagonalizable
Suppose A is a square matrix of size n with n distinct eigenvalues. Then A is diagonalizable.

Proof   Let λ1,λ2,λ3,,λn denote the n distinct eigenvalues of A. Then by Theorem NEM we have n = i=1nα A λi, which implies that αA λi = 1, 1 i n. From Theorem ME it follows that γA λi = 1, 1 i n. So γA λi = αA λi, 1 i n and Theorem DMFE says A is diagonalizable.

Example DEHD
Distinct eigenvalues, hence diagonalizable
In Example DEMS5 the matrix

H = 15 18 8 6 5 5 3 1 1 3 0 4 5 4 2 4346 17 14 15 26 30 12 8 10

has characteristic polynomial

pH x = x(x 2)(x 1)(x + 1)(x + 3)

and so is a 5 × 5 matrix with 5 distinct eigenvalues. By Theorem DED we know H must be diagonalizable. But just for practice, we exhibit the diagonalization itself. The matrix S contains eigenvectors of H as columns, one from each eigenspace, guaranteeing linear independent columns and thus the nonsingularity of S. The diagonal matrix has the eigenvalues of H in the same order that their respective eigenvectors appear as the columns of S. Notice that we are using the versions of the eigenvectors from Example DEMS5 that have integer entries.

S1HS = 2 1 1 1 1 1 0 2 0 1 2 0 2 12 41 0 21 2 2 1 2 1 1 15 18 8 6 5 5 3 1 1 3 0 4 5 4 2 4346 17 14 15 26 30 12 8 10 2 1 1 1 1 1 0 2 0 1 2 0 2 12 41 0 21 2 2 1 2 1 = 33 1 1 1 12 1 0 1 54 1 1 2 10103 2 4 76 1 1 3 15 18 8 6 5 5 3 1 1 3 0 4 5 4 2 4346 17 14 15 26 30 12 8 10 2 1 1 1 1 1 0 2 0 1 2 0 2 12 41 0 21 2 2 1 2 1 = 3 0 000 0 1000 0 0 000 0 0 010 0 0 002

Archetype B is another example of a matrix that has as many distinct eigenvalues as its size, and is hence diagonalizable by Theorem DED.

Powers of a diagonal matrix are easy to compute, and when a matrix is diagonalizable, it is almost as easy. We could state a theorem here perhaps, but we will settle instead for an example that makes the point just as well.

Example HPDM
High power of a diagonalizable matrix
Suppose that

A = 19 0 6 13 331 9 21 21 4 12 21 36 2 1428

and we wish to compute A20. Normally this would require 19 matrix multiplications, but since A is diagonalizable, we can simplify the computations substantially. First, we diagonalize A. With

S = 1 1 2 1 2 3 3 3 1 1 3 3 2 1 4 0

we find

D = S1AS = 6 1 36 0 2 23 3 0 1 2 11 1 1 19 0 6 13 331 9 21 21 4 12 21 36 2 1428 1 1 2 1 2 3 3 3 1 1 3 3 2 1 4 0 = 1000 0 000 0 020 0 001

Now we find an alternate expression for A20,

A20 = AAAA = InAInAInAInInAIn = SS1 A SS1 A SS1 A SS1 SS1 A SS1 = S S1AS S1AS S1AS S1ASS1 = SDDDDS1 = SD20S1  and since D is a diagonal matrix, powers are much easier to compute, = S 1000 0 000 0 020 0 001 20S1 = S (1)20 0 0 0 0 (0)20 0 0 0 0 (2)20 0 0 0 0 (1)20 S1 = 1 1 2 1 2 3 3 3 1 1 3 3 2 1 4 0 10 0 0 00 0 0 0010485760 00 0 1 6 1 36 0 2 23 3 0 1 2 11 1 1 = 6291451 2 2097148 4194297 9437175 531457196291441 9437175 2 3145728 6291453 12582900241942988388596

Notice how we effectively replaced the twentieth power of A by the twentieth power of D, and how a high power of a diagonal matrix is just a collection of powers of scalars on the diagonal. The price we pay for this simplification is the need to diagonalize the matrix (by computing eigenvalues and eigenvectors) and finding the inverse of the matrix of eigenvectors. And we still need to do two matrix products. But the higher the power, the greater the savings.

We close this section with a comment about an important theorem that we prove in one of the Topics sections. Every Hermitian matrix (Definition HM) is diagonalizable (Definition DZM), and the similarity transformation that accomplishes the diagonalization uses a unitary matrix (Definition UM). This means that for every Hermitian matrix of size n there is a basis of n that is composed entirely of eigenvectors for the matrix and also forms an orthonormal set (Definition ONS). Notice that for matrices with only real entries, we only need the hypothesis that the matrix is symmetric (Definition SYM) to reach this conclusion (Example ESMS4). Can you imagine a prettier basis for use with a matrix? I can’t. A precise statement of this result applies to a slightly broader class of matrices, known as “normal” matrices, see (link to definition here). With this expanded category of matrices, the result becomes an equivalence (Technique E). See (link to theorem here).

Subsection READ: Reading Questions

  1. What is an equivalence relation?
  2. State a condition that is equivalent to a matrix being diagonalizable, but is not the definition.
  3. Find a diagonal matrix similar to
    A = 58 47

Subsection EXC: Exercises

C20 Consider the matrix A below. First, show that A is diagonalizable by computing the geometric multiplicities of the eigenvalues and quoting the relevant theorem. Second, find a diagonal matrix D and a nonsingular matrix S so that S1AS = D. (See Exercise EE.C20 for some of the necessary computations.)

A = 1815 33 15 4 8 6 6 9 9 16 9 5 6 9 4

 
Contributed by Robert Beezer Solution [1246]

C21 Determine if the matrix A below is diagonalizable. If the matrix is diagonalizable, then find a diagonal matrix D that is similar to A, and provide the invertible matrix S that perfoms the similarity transformation. You should use your calculator to find the eigenvalues of the matrix, but try only using the row-reducing function of your calculator to assist with finding eigenvectors.

A = 1 9 9 24 3272968 1 11 13 26 1 7 7 18

 
Contributed by Robert Beezer Solution [1247]

C22 Consider the matrix A below. Find the eigenvalues of A using a calculator and use these to construct the characteristic polynomial of A, pA x. State the algebraic multiplicity of each eigenvalue. Find all of the eigenspaces for A by computing expressions for null spaces, only using your calculator to row-reduce matrices. State the geometric multiplicity of each eigenvalue. Is A diagonalizable? If not, explain why. If so, find a diagonal matrix D that is similar to A.

A = 19 25 30 5 2330355 7 9 10 1 3 4 5 1

 
Contributed by Robert Beezer Solution [1251]

T15 Suppose that A and B are similar matrices. Prove that A3 and B3 are similar matrices. Generalize.  
Contributed by Robert Beezer Solution [1253]

T16 Suppose that A and B are similar matrices, with A nonsingular. Prove that B is nonsingular, and that A1 is similar to B1.  
Contributed by Robert Beezer

T17 Suppose that B is a nonsingular matrix. Prove that AB is similar to BA.  
Contributed by Robert Beezer Solution [1254]

Subsection SOL: Solutions

C20 Contributed by Robert Beezer Statement [1243]
Using a calculator, we find that A has three distinct eigenvalues, λ = 3,2, 1, with λ = 2 having algebraic multiplicity two, αA 2 = 2. The eigenvalues λ = 3, 1 have algebraic multiplicity one, and so by Theorem ME we can conclude that their geometric multiplicities are one as well. Together with the computation of the geometric multiplicity of λ = 2 from Exercise EE.C20, we know

γA 3 = αA 3 = 1 γA 2 = αA 2 = 2 γA 1 = αA 1 = 1

This satisfies the hypotheses of Theorem DMFE, and so we can conclude that A is diagonalizable.

A calculator will give us four eigenvectors of A, the two for λ = 2 being linearly independent presumably. Or, by hand, we could find basis vectors for the three eigenspaces. For λ = 3, 1 the eigenspaces have dimension one, and so any eigenvector for these eigenvalues will be multiples of the ones we use below. For λ = 2 there are many different bases for the eigenspace, so your answer could vary. Our eigenvectors are the basis vectors we would have obtained if we had actually constructed a basis in Exercise EE.C20 rather than just computing the dimension.

By the construction in the proof of Theorem DC, the required matrix S has columns that are four linearly independent eigenvectors of A and the diagonal matrix has the eigenvalues on the diagonal (in the same order as the eigenvectors in S). Here are the pieces, “doing” the diagonalization,

1 0 3 6 211 0 0 0 1 3 1 1 0 1 1 1815 33 15 4 8 6 6 9 9 16 9 5 6 9 4 1 0 3 6 211 0 0 0 1 3 1 1 0 1 = 300 0 020 0 002 0 0001

C21 Contributed by Robert Beezer Statement [1243]
A calculator will provide the eigenvalues λ = 2,2,1,0, so we can reconstruct the characteristic polynomial as

pA x = (x 2)2(x 1)x

so the algebraic multiplicities of the eigenvalues are

αA 2 = 2 αA 1 = 1 αA 0 = 1

Now compute eigenspaces by hand, obtaining null spaces for each of the three eigenvalues by constructing the correct singular matrix (Theorem EMNS),

A 2I4 = 1 9 9 24 3292968 1 11 11 26 1 7 7 16  RREF 1003 2 011 5 2 000 0 000 0 A 2 = NA 2I4 = 3 2 5 2 0 1 , 0 1 1 0 = 3 5 0 2 , 0 1 1 0 A 1I4 = 0 9 9 24 3282968 1 11 12 26 1 7 7 17  RREF 1005 3 010 13 3 0015 3 000 0 A 1 = NA I4 = 5 3 13 3 5 3 1 = 5 13 5 3 A 0I4 = 1 9 9 24 3272968 1 11 13 26 1 7 7 18  RREF 1003 010 5 0012 000 0 A 0 = NA I4 = 3 5 2 1

From this we can compute the dimensions of the eigenspaces to obtain the geometric multiplicities,

γA 2 = 2 γA 1 = 1 γA 0 = 1

For each eigenvalue, the algebraic and geometric multiplicities are equal and so by Theorem DMFE we now know that A is diagonalizable. The construction in Theorem DC suggests we form a matrix whose columns are eigenvectors of A

S = 3 0 5 3 51135 0 1 5 2 2 0 3 1

Since det S = 10, we know that S is nonsingular (Theorem SMZD), so the columns of S are a set of 4 linearly independent eigenvectors of A. By the proof of Theorem SMZD we know

S1AS = 2000 0200 0010 0000

a diagonal matrix with the eigenvalues of A along the diagonal, in the same order as the associated eigenvectors appear as columns of S.

C22 Contributed by Robert Beezer Statement [1244]
A calculator will report λ = 0 as an eigenvalue of algebraic multiplicity of 2, and λ = 1 as an eigenvalue of algebraic multiplicity 2 as well. Since eigenvalues are roots of the characteristic polynomial (Theorem EMRCP) we have the factored version

pA x = (x 0)2(x (1))2 = x2(x2 + 2x + 1) = x4 + 2x3 + x2

The eigenspaces are then

λ = 0 A (0)I4 = 19 25 30 5 2330355 7 9 10 1 3 4 5 1  RREF 1055 01 5 4 00 0 0 00 0 0 A 0 = NC (0)I4 = 5 5 1 0 , 5 4 0 1

λ = 1 A (1)I4 = 20 25 30 5 2329355 7 9 11 1 3 4 5 0  RREF 101 4 01 2 3 00 0 0 00 0 0 A 1 = NC (1)I4 = 1 2 1 0 , 4 3 0 1 Each eigenspace above is described by a spanning set obtained through an application of Theorem BNS and so is a basis for the eigenspace. In each case the dimension, and therefore the geometric multiplicity, is 2.

For each of the two eigenvalues, the algebraic and geometric multiplicities are equal. Theorem DMFE says that in this situation the matrix is diagonalizable. We know from Theorem DC that when we diagonalize A the diagonal matrix will have the eigenvalues of A on the diagonal (in some order). So we can claim that

D = 00 0 0 00 0 0 001 0 00 0 1

T15 Contributed by Robert Beezer Statement [1245]
By Definition SIM we know that there is a nonsingular matrix S so that A = S1BS. Then

A3 = (S1BS)3 = (S1BS)(S1BS)(S1BS) = S1B(SS1)B(SS1)BS  Theorem MMA = S1B(I 3)B(I3)BS  Definition MI = S1BBBS  Theorem MMIM = S1B3S

This equation says that A3 is similar to B3 (via the matrix S).

More generally, if A is similar to B, and m is a non-negative integer, then Am is similar to Bm. This can be proved using induction (Technique I).

T17 Contributed by Robert Beezer Statement [1245]
The nonsingular (invertible) matrix B will provide the desired similarity transformation,

B1 BAB = B1B AB  Theorem MMA = InAB  Definition MI = AB  Theorem MMIM