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Realm of Matrices Exponential and Logarithm Functions 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 orthog- onal 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 transforma- tions 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, uni- potent, 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 prop- erties [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. RESONANCE ⎜ February 2015 137 GENERAL ⎜ ARTICLE 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. 138 RESONANCE ⎜ February 2015 GENERAL ⎜ ARTICLE 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. RESONANCE ⎜ February 2015 139 GENERAL ⎜ ARTICLE which has ones along the super diagonal and zeros at other places. As a linear transformation, this shift ma- trix 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. 140 RESONANCE ⎜ February 2015 GENERAL ⎜ ARTICLE 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 ). RESONANCE ⎜ February 2015 141 GENERAL ⎜ ARTICLE − 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.
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