Realm of Matrices Exponential and Logarithm Functions

Home , Diagonalizable matrix, Diagonal matrix, Matrix exponential, Nilpotent, Nilpotent matrix, Orthogonal matrix

GENERAL ⎜ ARTICLE Realm of Matrices Exponential and Logarithm Functions

Debapriya Biswas

In this article, we discuss the exponential and the logarithmic functions in the realm of matrices. These notions are very useful in the mathemat- ical and the physical sciences [1,2]. We discuss some important results including the connections established between skew-symmetric and orthogonal matrices, etc., through the exponential map. Debapriya Biswas is an Assistant Professor at the 1. Introduction Department of Mathemat- ics, IIT- Kharagpur, West The term ‘matrix’ was coined by Sylvester in 1850. Car- Bengal, India. Her areas of dano, Leibniz, Seki, Cayley, Jordan, Gauss, Cramer and interest are Lie groups and Lie algebras and their others have made deep contributions to matrix theory. representation theory, The theory of matrices is a fundamental tool widely used harmonic analysis and in different branches of science and engineering such as complex analysis, in classical mechanics, optics, electromagnetism, quantum particular, Clifford mechanics, motion of rigid bodies, astrophysics, prob- analysis. She is interested in teaching also and enjoys ability theory, and computer graphics [3–5]. The stan- discussing her research dard way that matrix theory gets applied is by its role as interests with others. a representation of linear transformations and in finding solutions to a system of linear equations [6]. Matrix al- gebra describes not only the study of linear transformations and operators, but it also gives an insight into the geometry of linear transformations [7]. Matrix calculus generalizes the classical analytical notions like deriva- tives to higher dimensions [8]. Also, infinite matrices (which may have an infinite number of rows or columns) occur in planetary theory and atomic theory. Further, the classification of matrices into different types such Keywords as skew-symmetric, orthogonal, nilpotent, or unipotent Matrix exponential, matrix loga- matrices, is essential in dealing with complicated practi- rithm, orthogonal, nilpotent, unipotent, skew-symmetric, Jordan cal problems. In this article, we will discuss the method matrix. to compute the exponential of any arbitrary real or com-

136 RESONANCE ⎜ February 2015 GENERAL ⎜ ARTICLE plex matrix, and discuss some of their important properties [9,10]. 2. Jordan Form of Matrices A Jordan block – named in honour of Camille Jordan – is a matrix of the form ⎛ ⎞ λ 10... 0 ⎜ ⎟ ⎜0 λ 1 ... 0⎟ ⎜ ⎟ ⎜ ...... ⎟ J = ⎜ . . . . . ⎟ . ⎝00... λ 1⎠ 00... 0 λ

Every Jordan block that is described by its dimension n and its eigenvalue λ, is denoted by Jλ,n. DEFINITION 2.1

If Mn denotes the set of all n×n complex matrices, then a matrix A ∈ Mn of the form ⎛ ⎞ A11 0 ⎜ ⎟ ⎜ A22 ⎟ A = ⎜ . ⎟ ⎝ .. ⎠ 0 Akk

in which Aii ∈ Mni ,i =1, 2,...,k,andn1 + n2 + ...+ nk = n, is called a block diagonal. Notationally, such a matrix is often indicated as A = A11⊕A22⊕...⊕Akk;this is called the direct sum of the matrices A11,A22,...,Akk [7]. A block diagonal matrix whose blocks are Jordan blocks, is called a Jordan matrix, denoted by using either ⊕ or diag symbol. A block diagonal The (m+s+p)×(m+s+p) block diagonal square matrix, matrix whose having ﬁrst, second, and third diagonal blocks Ja,m,Jb,s blocks are Jordan and Jc,p is compactly indicated as Ja,m ⊕ Jb,s ⊕ Jc,p or, blocks, is called a diag(Ja,m,Jb,s,Jc,p) respectively [4,7]. For example, the Jordan matrix.

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square matrix ⎛ ⎞ 0100000000 ⎜ ⎟ ⎜0010000000⎟ ⎜ ⎟ ⎜0000000000⎟ ⎜ ⎟ ⎜000i 100000⎟ ⎜ ⎟ ⎜0000i 00000⎟ J = ⎜ ⎟ ⎜00000i 1000⎟ ⎜ ⎟ ⎜000000i 000⎟ ⎜ ⎟ ⎜0000000510⎟ ⎝0000000051⎠ 0000000005 is a 10 × 10 Jordan matrix with a 3 × 3blockwith eigenvalue 0, two 2 × 2 blocks with imaginary unit i and a 3 × 3 block with eigenvalue 5. Its Jordan block structure can be expressed as either J0,3 ⊕Ji,2 ⊕Ji,2 ⊕J5,3 or, diag(J0,3,Ji,2,Ji,2,J5,3). 3. Nilpotent and Unipotent Matrices DEFINITION 3.1 A square matrix X is said to be nilpotent if Xr =0for some positive integer r. The least such positive integer is called the index (or, degree) of nilpotency. If X is an n × n nilpotent matrix, then Xm =0forallm ≥ n [9]. × 0 2 For example, the 2 2matrixA =(0 0 ) is nilpotent of degree 2, since A2 = 0. In general, any triangular matrix with zeros along the main diagonal is nilpotent. For example, the 4 × 4matrix ⎛ ⎞ 0124 ⎜ ⎟ 0021⎟ A = ⎜ ⎝0005⎠ In general, any 0000 triangular matrix with zeros along is nilpotent of degree 4 as A4 =0andA3 =0.Inthe the main diagonal above examples, several entries are zero. However, this is nilpotent. may not be so in a typical nilpotent matrix.

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For instance, the 3 × 3 matrix ⎛ ⎞ 5 −32 A = ⎝15 −96⎠ 10 −64 squares to zero, i.e., A2 = 0, though the matrix has no zero entries.

For A ∈ Mn, the following characterization may be worth mentioning:

• Matrix A is nilpotent of degree r ≤ n i.e., Ar =0.

• The characteristic polynomial χA(λ)=det(λIn − A)ofA is λn. • The minimal polynomial for A is λr. • tr(Ar) = 0 for all r>0, i.e., the sum of all the diagonal entries of Ar vanishes. • The only (complex) eigenvalue of A is 0.

Further, from the above, the following observations can be added:

• The degree of an n × n nilpotent matrix is always less than or equal to n. • The determinant and trace of a nilpotent matrix are always zero. • The only nilpotent diagonalizable matrix is the zero matrix.

3.2 Canonical Nilpotent Matrix × We consider the n ⎛n shift matrix ⎞ 010... 0 The only nilpotent ⎜ ⎟ ⎜001... 0⎟ diagonalizable A = ⎜. . . . ⎟ ⎝. . . .. 1⎠ matrix is the zero 000... 0 matrix.

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which has ones along the super diagonal and zeros at other places. As a linear transformation, this shift matrix shifts the components of a vector one slot to the n left: S(a1,a2,...,an)=(a2,a3,...,an, 0). As, A = 0 = An−1,thismatrixA is nilpotent of degree n and is called the canonical nilpotent matrix. Further, if A is any nilpotent matrix, then A is similar to a block diagonal matrix of the form ⎛ ⎞ A1 O ... O ⎜ ⎟ ⎜ OA2 ... O⎟ ⎜ . . . ⎟ , ⎝ . . .. O ⎠ O O ... Ar

where each of the blocks A1,A2,...,Ar is a shift matrix (possibly of diﬀerent sizes). The above theorem is a special case of the Jordan canonical form of matrices. For example, any non-zero, nilpotent, 2-by-2 matrix A is 0 1 similar to the matrix ( 0 0 ). That is, if A is any non-zero nilpotent matrix, then there exists a basis {b1,b2} such 0 1 that Ab1 = O and Ab2 = b1. For example, if A =(0 0 ), 1 0 0 1 b1 =(0 ), b2 =(1 ), then Ab1 =(0 )andAb2 =(0 )=b1.

3.3 Properties

(i) If A is a nilpotent matrix, then I + A is invertible. Moreover, (I + A)−1 = I − A + A2 − A3 + ···+ (−1)n−1An−1, where the degree of A is n.

(ii) If A is nilpotent then det(I +A)=1.Forexample, 0 1 2 if A =(0 0 ), then, A = O and det(I + A)=1. Conversely, if A is a matrix and det(I + tA)=1 Every singular for all values of scalar t then A is nilpotent. matrix can be (iii) Every singular matrix can be expressed as a prod- expressed as a uct of nilpotent matrices. product of nilpotent matrices.

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DEFINITION 3.4. An n × n matrix A is said to be An n × n matrix A is said to be unipotent if the matrix A − I is nilpotent. The degree of nilpotency of A − I is unipotent if the also called the degree of unipotency of A. matrix A – I is nilpotent. For example, ⎛ ⎞ 1124 ⎜ ⎟ 13 ⎜0121⎟ A = ,B = 01 ⎝0015⎠ 0001 and ⎛ ⎞ 6 −32 C = ⎝15 −86⎠ 10 −65 are unipotent matrices of degree 2, 4 and 2 respectively because (A − I)2 = O,(B − I)4 = O and (C − I)2 = O. We know that every complex matrix X is similar to an upper triangular matrix. Thus, there exists a non- singular matrix P such that X = PAP−1,where ⎛ ⎞ a11 a12 ... a1n ⎜ ⎟ ⎜ 0 a22 ... a2n⎟ A = ⎜ . . . . ⎟ . ⎝ . . .. . ⎠ 00... ann

Therefore, the characteristic polynomial of X is (λ − a11)(λ − a22) ...(λ − ann), as similar matrices have the same characteristic polynomial. Then two cases may arise:

Case I. The eigenvalues a11,a22,...,ann are all distinct.

Case II. Not all of a11,a22,...,ann are distinct.

3 2 Forexample,considerthematrixA =(1 4 ).

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− 3−λ 2 − − − Then A λI =( 1 4−λ )anddet(A λI)=(λ 5)(λ 2). The determinant vanishes if λ = 5 or 2 which are the distinct eigenvalues of A. Now to find the eigenvectors of the matrix equation AX = λX, we solve the two systems of linear equations (A − 5I)X =0and(A − 2I)X =0 1 where from the eigen vectors are obtained as v1 =(1 ) 2 and v2 =(−1 ). 2 These eigenvectors form a basis B =(v1,v2)ofR and the matrix relating the standard basis E to the basis B −1 1 2 −1 − 1 −1 −2 −1 is P =(B) =(1 −1 ) = 3 ( −1 1 )andPAP = A − 1 −1 −2 3 2 1 2 5 0 is diagonal: A = 3 ( −1 1 )(1 4 )(1 −1 )=(0 2 ), which is the Jordan canonical form of A, and its characteristic polynomial is (λ−5)(λ−2). The two distinct eigenvalues are 5 and 2. 4. Exponential of a Matrix z zn Recall that the exponential function e = n! is a n≥0 convergent series for each z ∈ C. Before we explain the meaning of the above infinite series when z is replaced by a general n × n matrix, we describe it for the case of nilpotent matrices. If X is a nilpotent matrix of degree r, then by definition Xr =0,r ≤ n,sothat

n 2 3 r−1 X X X X X X e = = I + + + + ...+ − n≥0 n! 1! 2! 3! (r 1)!

is a polynomial in X of degree (r − 1). Forexample,considerthe4× 4 matrix ⎛ ⎞ 0124 ⎜ ⎟ ⎜0021⎟ A = . ⎝0005⎠ 0000

By direct multiplication

142 RESONANCE ⎜ February 2015 GENERAL ⎜ ARTICLE ⎛ ⎞ 00211 ⎜ ⎟ 2 00010 A = ⎜ ⎟ , ⎝000 0⎠ 000 0 ⎛ ⎞ 00010 ⎜ ⎟ 3 ⎜000 0⎟ A = , ⎝000 0⎠ 000 0 and A4 = 0; therefore, A is a nilpotent matrix of degree 4. The exponential series for this matrix A reduces to A 1 2 1 3 e = I +A+ 2 A + 6 A and by matrix addition, it yields a4× 4 matrix ⎛ ⎞ 113 67 ⎜ 6 ⎟ A ⎜012 6⎟ e = ⎝001 5⎠ 000 1 which is invertible since det(eA)=1. To explain the meaning of the power series for a general matrix, deﬁne the norm of any A ∈ Mn to be

||A|| =Maxi,j|aij|. Note that it is easy to check ||AB|| ≤ n||A||||B||. The norm gives a notion of distance between matrices viz., the distance between A and B is the norm of A−B. With this deﬁnition, it follows that k r n||A|| A e − 1 || || ≤ 1+ ∀ k. r=0 r! n k Ar Therefore, in Mn, the sequence of matrices r=0 r! converges to a matrix as k →∞; this matrix is denoted by eA. Here are some properties of the exponential of a matrix:

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On det(eA) = etr(A) • e = In,whereOn is the n × n zero matrix.

for any A∈M . −1 n • PeAP −1 = ePAP . • If B is upper triangular, then eB is upper triangular with diagonal entries ebii . In particular, since every matrix is similar to an upper triangular matrix, we have

det(eA)=etr(A)

for any A ∈ Mn.

5. Logarithm of Unipotent and other Matrices − x2 x3 −··· | | Recall that log(1 + x)=x 2 + 3 for x < 1. If a matrix A is nilpotent of degree r then Ar = O for a positive constant r, and it makes sense to deﬁne A2 A3 log(I + A)=A − + − ... 2 3 r−1 r−2 A +(−1) . (r − 1) On the other hand, if the matrix A is unipotent, then A−I is nilpotent so that (A−I)r = O for some positive constant r.LetN = A − I. Therefore, (A − I)2 (A − I)3 log A =(A − I) − + − ... 2 3 r−1 r−2 (A − I) +(−1) . r − 1 Forexample,considerthematrix ⎛ ⎞ 1124 ⎜0121⎟ A = ⎜ ⎟ ⎝0015⎠ 0001

so that A−I,(A−I)2, (A−I)3 are non-zero matrices and − 4 − − (A−I)2 (A−I)3 (A I) = O. Then, log A =(A I) 2 + 3 ,

144 RESONANCE ⎜ February 2015 GENERAL ⎜ ARTICLE becomes ⎛ ⎞ The fundamental 011 11 ⎜ 6 ⎟ property of the ⎜002−4⎟ log A = ⎝ ⎠ matrix logarithm is 000 5 the same as that 000 0 of ordinary which is the logarithm of the unipotent matrix A of de- logarithm; it is the gree 4. inverse of exponential More generally, we may deﬁne: function. DEFINITION 5.1. The logarithm of a square matrix I + A with ||A|| < 1 is deﬁned by [10]

A2 A3 A4 log(I + A)=A − + − + ... . 2 3 4 This series is convergent for ||A|| < 1, and hence log(I + A) is a well-deﬁned continuous function in this neigh- borhood of I. The fundamental property of the matrix logarithm is the same as that of ordinary logarithm; it is the inverse of exponential function [10], i.e., log A = B implies eB = A. Lemma 5.2. If an n × n matrix A is unipotent, then log A is nilpotent. Proof. We know that for any n × n matrix A,

∞ m (A − I) log A = (−1)m+1 (1) m=1 m whenever the series converges. Since A is unipotent, then A − I is nilpotent. Let the degree of nilpotency of (A−I)ber.Then(A−I)m = O for all m ≥ r. So the series (1) becomes

r−1 m (A − I) log A = (−1)m+1 , (2) m=1 m

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i.e., the series terminates. So log A is deﬁned whenever A is unipotent. Let A−I = N.ThenA = I+N and N r =(A−I)r = O. Therefore from (2),

2 3 r−1 N N r N log A = N − + −···(−1) , 2 3 r − 1 and so 2 3 r−1 r r − N N −··· − r N (log A) = N + ( 1) − 2 3 r 1 2 r−2 r r N N r N = N I − + −···(−1) 2 3 r − 1 = O.

Therefore log A is nilpotent of degree r. PROPOSITION 5.3. If a matrix A is unipotent, then exp(log A)=A. Proof.SinceA is unipotent, then A = I + N,where N is nilpotent. As such from the above lemma, log A is deﬁned. Let the degree of nilpotency of N be r,thatis, N m = O for all m ≥ r.Therefore, exp(log A)

=explog( I + N) 2 3 r−1 − N N −··· − r N =exp N + +( 1) − 2 3 r 1 2 3 r−1 N N r N = I + N − + −···+(−1) 2 3 r − 1 2 3 r−1 2 1 − N N −··· − r N ··· + N + +( 1) − + 2! 2 3 r 1 1 N 2 N 3 + N − + −··· If a matrix A is (r − 1)! 2 3 unipotent, then r−1 r−1 r N +(−1) = I + N, exp(log A) = A. r − 1

146 RESONANCE ⎜ February 2015 GENERAL ⎜ ARTICLE as the coeﬃcients of N 2,N3,...,Nr−1 are all zero. Hence, exp(log A)=I + N = A. PROPOSITION 5.4 It may be shown that

d −1 log(I + At)=A(I + At) . dt

Proof. By deﬁnition of logarithmic series, d d (At)2 (At)3 log(I + At)= At − + − ... dt dt 2 3 − 2 3 2 − = A A t + A t ... 2 = A I − (At)+(At) − ... −1 = A(I + At) .

6. Properties of Exponential Map on Matrices We have already mentioned some properties (the ﬁrst three below) of the exponential map on matrices. Let us recall some more.

On • e = In,whereOn is the n × n zero matrix.

−1 • PeAP −1 = ePAP .

• If B is upper triangular, then eB is upper triangular with diagonal entries ebii . In particular, since every matrix is similar to an upper triangular matrix, we have

det(eA)=etr(A)

for any A ∈ Mn.

• If A, B ∈ Mn commute (that is, AB = BA), then eA+B = eAeB.

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If A is any n × n • If A is any n × n complex matrix, then the expo- A −A complex matrix, nential e is invertible and its inverse is e . then the exponential eA is We point out the following examples for computing the invertible and its exponential map: inverse is e–A. × 1 1 (1) Consider a triangular 2 2 matrix A =(0 2 ). Then 2 13 3 17 4 115 A = ,A = ,A = , ··· , 04 08 016

so that 2 3 4 A A A A e = I + A + + + + ... 2! 3! 4! ee2 − e = 2 . 0 e

(2) The above matrix eA is a non-singular matrix and its inverse is computed as −1 −2 −1 −A A −1 e e − e e =(e ) = −2 . 0 e

× 2 3 (3) Consider a 2 2 real matrix A =(0 2 )whichmay 1 0 be expressed as A =2I + B,whereI =(0 1 )and 0 3 B =(0 0 ). Since 2I commutes with B,wecan write eA = e2I+B = e2I eB. From the exponential 2I 2 B 1 3 series, we ﬁnd e = e I,ande = I + B =(0 1 ) as B2 = 0. Therefore, 2 2 2 A e 0 13 e 3e e = 2 = 2 0 e 01 0 e which is a non-singular matrix with inverse e−A = e−2 −3e−2 ( 0 e−2 ).

× d1 0 (4) Consider a 2 2 diagonal matrix A =(0 d2 ).

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Then d1 A A2 A3 A4 e 0 e = I + A + + + + ...= d . 2! 3! 4! 0 e 2 −A e−d1 0 Its inverse is computed as e =( 0 e−d2 ).

7. Relation of Skew-symmetric Matrices to Or- thogonal Matrices DEFINITION 7.1 An n × n matrix A satisfying A + At = 0, is called a skew-symmetric matrix. That is, its transpose is equal to the negative of itself, i.e., At = −A. The name skew-symmetric comes because the reﬂection of each entry about the diagonal is the negative of that entry. In particular, all leading diagonal elements are zero, i.e., tr(A)=0. DEFINITION 7.2 An n × n matrix A satisfying AAt = I, is called an orthogonal matrix. That is, its transpose equals its inverse, At = A−1,[5]. For examples, the following are skew-symmetric matrices: ⎛ ⎞ 0 −hg 01 A = ,B = ⎝ h 0 −f⎠ , −10 −gf 0

⎛ ⎞ 0 −12−3 ⎜ ⎟ ⎜ 10−45⎟ C = . ⎝−24 0−6⎠ 3 −56 0 Some examples of orthogonal matrices are: −10 cos θ sin θ A = ,B = . 0 −1 − sin θ cos θ

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The addition formula of the exponential function is eA+B = eAeB,whenAB = BA.IfX is skew-symmetric, then the matrices X and Xt commute, as X + Xt = O.That is, XXt = X(−X)=(−X)X = XtX,whereXt is the transpose of X. Thus, it follows from the above property t t of exponential function that eXeX = eX+X = eO = I, where O and I are zero and identity matrices. Also, t t eX =(eX)t. Therefore, I = eX eX = eX (eX)t.That is, if X is skew-symmetric, then eX is an orthogonal matrix. Further, eX has determinant 1, as det(eX)= etr(x) = e0 =1. Finally, we leave as an exercise the connection between skew-hermitian matrices and unitary matrices. The exponential of a matrix plays a very important role in the theory of Lie groups.