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Matrix Algebra and Control Appendix A Matrix Algebra and Control Boldface lower case letters, e.g., a or b, denote vectors, boldface capital letters, e.g., A, M, denote matrices. A vector is a column matrix. Containing m elements (entries) it is referred to as an m-vector. The number of rows and columns of a matrix A is nand m, respectively. Then, A is an (n, m)-matrix or n x m-matrix (dimension n x m). The matrix A is called positive or non-negative if A>, 0 or A :2:, 0 , respectively, i.e., if the elements are real, positive and non-negative, respectively. A.1 Matrix Multiplication Two matrices A and B may only be multiplied, C = AB , if they are conformable. A has size n x m, B m x r, C n x r. Two matrices are conformable for multiplication if the number m of columns of the first matrix A equals the number m of rows of the second matrix B. Kronecker matrix products do not require conformable multiplicands. The elements or entries of the matrices are related as follows Cij = 2::;;'=1 AivBvj 'Vi = 1 ... n, j = 1 ... r . The jth column vector C,j of the matrix C as denoted in Eq.(A.I3) can be calculated from the columns Av and the entries BVj by the following relation; the jth row C j ' from rows Bv. and Ajv: column C j = L A,vBvj , row Cj ' = (CT),j = LAjvBv, (A. 1) /1=1 11=1 A matrix product, e.g., AB = (c: c:) (~b ~b) = 0 , may be zero although neither multipli­ cand A nor multiplicator B is zero. Without A or B being the nullmatrix the product AB only vanishes if both A and B are singular. The matrix B = A 1. is the (right) annihilator of A, i.e., AA 1. = O. A.2 Properties of Matrix Operations Distributivity: A(B + C) = AB + AC . Associativity of addition (A + B) + C = A + (B + C) and multiplication (AB)C = A(BC). Commutativity of addition, non-commutativity of multiplication and raise to higher powers: A+B=B+A, AB f- BA, (A.2) Exceptions: Consider, first, a multi variable control with transfer matrix G( s) in the forward path and unity feedback and, second, H in the forward path and F in the feedback where G = FH. The overall transfer matrix is given by (I + FH)-IFH = FH(I + FH)-I . (A.3) The inverse of the return-difference matrix and G commute unexpectedly. Another exceptional case is the product A exp (At), i.e. the coefficient matrix and the state transition matrix. Finally, suppose A and B nonsingular. Then, A and B commute if their product is the identity matrix: Both AB = I and BA = I yield A = B-1 . Generally, the matrices A and B commute with respect to multiplication if B is a function of A , e.g. as given by a matrix polynomial or by the decomposition in Eq.(A.45). 610 A Matrix Algebra and Control Note that F(I + HF)-I = (I + FH)-I F and F(I + HF)-I H = (I + FH)-IFH , particularly observe the change of order within the parentheses. 0 Properties when transposing or inverting a matrix product: (AA) Inverse and transpose operations (symbols) may be permuted: (AT)-I = (A-If. If A-I = AT is true then A is referred to as an orthogonal matrix. An idempotent matrix A has the property A 2 = A . This result can be observed in least squares, estimation and sliding mode theory, e.g., or A = I - B(C,B)-IC, . (A.5) A matrix A is nilpotent if A k = 0 for some k. Such a matrix appears in the case of the state-space representation of a k-tuples integrator. A.3 Diagonal Matrices A diagonal matrix A is a square matrix with non-zero entries A;; in the main diagonal, only, e.g., (A.6) If these entries A;; are equal to each other A is a scalar matrix. The identity matrix Inis a scalar matrix with elements 1 and dimension n x n: In ~ dia~(1,I,I, ... ,1), In E nnxn . Given a rectangular (n, r)-matrix B, premultiplying B by the identity matrix In or postmultiplying B by Ir yields InB = B or Blr = B . Premultiplying [postmultiplying] a matrix A by a diagonal matrix yields ... ... ... ) ) , (A.7) i.e., a new matrix the rows [columns] of which are successively multiplied (scaled) by d; (i.e., i-th row [column] with di). A.4 Triangular Matrices A lower triangular matrix is a square matrix having all elements zero above the main diagonal, an upper triangular matrix only contains zero elements below the main diagonal. The product of two triangular matrices produces a triangular one again. If A is given as a diagonal matrix or an upper or lower triangular matrix, the eigenvalues A[A] are already given by the entries in the main diagonal A;; Vi = L.n. A.S Column Matrices (Vectors) and Row Matrices The unit m-vector with k-th component 1 is termed ek . Defining this m-vector ek and the n-vector ei , the elementary matrix or Kronecker matrix E;k is given by the dyadic product ek = (0,0, ... I, ... ,0, of = e~mx I) = (Im).k E;k E nnxm (A.8) as an (n,m)-matrix with entry 1 only in the i,k-element and zero elsewhere. Thus, the (n,m)-matrix A can be established element by element: A = L7=1 L;'=I A;kEik where A E nnxm or dim A = n X m . The identity matrix can be achieved by the sum In = L7=1 e;er = L7=1 E;; . The sum vector with elements throughout unity 1 = (I, 1, ... ,If serves as a summation operator for an m-vector: ITa= L~ai. A.6 Reduced Matrix, Minor, Cofactor, Adjoint 611 The inner product of two vectors a and b is a scalar aTb = bT a . Orthogonal vectors have zero inner product. Assume a vector output signal y given by the linear combination of a vector input signal x governed by the transfer equation y=Cx where yEnr , xEnn , CEnrxn . (A.9) The entry Gij of C is considered as an operational factor from the input component Xj to the output component Yi. Note that the output subscript i (effect) is written first and j (source) is written second. A partitioned vector u is denoted by the "vec" symbol u- (:~ ) =vec(uI,u2, ... uN)=vecui=(uf,uI, ... u~f uiEnm " uEnm , m=Lmi. - u~ (A.lO) The norm or length of a is the distance to the nullvector (origin) and is defined by the Frobenius norm lIaliF = .,;;T; . The inner product is always smaller than the product of the norms of the multiplicators (Schwartz inequality): laTbl ::; lIallFllbliF . Triangle inequality (for any kind of norm): lIa + bll ::; Iiall + IIbll . The angle 0 between two vectors u and v is defined by cosO = uTv/(lluIlFllvIlF) . Mapping a matrix A (A.ll) to a vector a is provided by the operator "col" (or by the operator "vec") colA = vecA = a = (All ... AmI: AI2 ... Am2 ... Amnf (A.I2) The operator col lists the entries of A column-wise. Separating the ith column of A will be termed by (A).i. For abbreviation also A.i is used although, usually, only matrices are denoted by upper case boldface letters, irrespective of the subscript. A.i is defined as a column matrix (vector). The operator col (column string) can also be written as col A = (A·f : A·I ... A.~f. (A.I3) The inner product of two real matrices A and B of equal dimension m x n is a scalar and coincides with the trace of the matrix product: (A, B) = (coIA)T colB = trATB = L:::I L:j=1 AijBij ::; IIAIIFIIBIIF . The Frobenius or Euclidian norm of a real matrix A is given by (A.I4) A.6 Reduced Matrix, Minor, Cofactor, Adjoint Given the (n, n)-matrix A , the reduced matrix A"d ik of the size (n - 1) x (n - 1) is obtained by cancelling row i and column k. Repeating for NI ... n, k'v'I ... n yields n 2 different matrices. The minor on the i, k-component of A is defined as the determinant det Ared ik . The cofactor of the i, k-element of A is obtained by permuting the sign of the minor, precisely by multiplying the minor with (-1 )iH, that is, COfikA = (_1)iH det Ared ik . Given the (n, n)-matrix A on the elements Aik, then A = (Aik) = matrix[AikJ and the adjoint is given as the transposed matrix of cofactors: adjA = [matrix( COfikA)JT . The determinant can be decomposed with respect to the row i , or with respect to the kth column, that is, n det A = L AikcofikA Vi = 1 ... n or det A = L AikcofikA 'v'k=l. .. n. (A.I5) k=1 i=l Interpretation of the equations above as a matrix multiplication yields I det A = A adjA = (adjA) A A -I = adjA . (A.I6) detA 612 A Ma.trix Algebra. and Control A.7 Similar Matrices Matrices A and A are similar, i.e., A ~ A, if A = TAT-I. The preceding equation is named similarity transformation. Similar matrices A and A are characterized by the property det A = det A, by the same eigenvalues, eigenvalue multiplicities and eigenvalue indices (and by the same number of generalized eigenvectors), see Eq.(B.I07). Examples of similar matrices are A and A = diag ~i[A) = T-I AT in Eq.(B.I3).
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