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B4 MATRIX INVERSION is a scalar real number that can be computed either as in Software
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Matrix inversion is important is both parameter and state estimation Often problems can be manipulated into the form y = A where y and A are known, but is unknown When A is square and nonsingular, then its inverse exists and the unique solution is = A 1 y For A Rm n with m > n and rank(A) = n, then the solution = A A minimizes the norm of the error vector (y A ) This least squares solution is derived in Section 532 When A is a square nonsingular matrix, A 1 = C A (B29) where C is the cofactor matrix for A and C is called the adjoint of A The matrix inverse has the following properties: 1 A 1 2 (AB) 1 1 = A, = B 1 A 1 , 1 A , 3 A 1 = 4 A 5 ( A) 1 1 = A 1 = 1 1 A which will be denoted by A , and
The inverse of an orthonormal matrix is the same as its transpose
Matrix Inversion Lemma
Two forms of the Matrix Inversion Lemma are presented The Lemma is useful in least squares and Kalman lter derivations Each lemma can be proved by direct multiplication Lemma B51 Given four matrices P1 , P2 , H, and R of compatible dimensions, if P1 , P2 , R, and H P1 H + R are all invertible and P 1 = P1 1 + HR 1 H , 2 then P 2 = P 1 P1 H H P 1 H + R (B30) H P1 (B31) B6 EIGENVALUES AND EIGENVECTORS
Lemma B52 Given four matrices A, B, C, and D of compatible dimensions, if A, C, and A + BCD are invertible, then (A + BCD) = A 1 A 1 B DA 1 B + C 1 DA 1
(B32) The equivalence of the two forms is shown by de ning: A = P 1 , 1 B = H, C = R 1 , D=H , and requiring A + BCD = P 1 2
Eigenvalues and Eigenvectors
For A Rn n , the set of scalars i C and (nonzero) vectors xi C n satisfying or ( i I A)xi = 0n Axi = i xi are the eigenvalues and eigenvectors of A We are only interested in nontrivial solutions (ie, solution xi = 0 is not of interest) Nontrivial solutions exist only if ( i I A) is a singular matrix Therefore, the eigenvalues of A are the values of such that  I A = 0 This yields an nth order polynomial in If A is a symmetric matrix, then all of its eigenvalues and eigenvectors are real If xi and xj are eigenvectors of symmetric matrix A and their eigenvalues are not equal (ie, i = j ), then the eigenvectors are orthogonal (ie xi xj = 0) A square matrix A is idempotent if and only if AA = A Idempotent matrices are sometimes also called projection matrices Idempotent matrices have the following properties: 1 rank(A) = T r(A); 2 the eigenvalues of A are all either 0 or 1; 3 the multiplicity of 1 as an eigenvalue is the rank(A); 4 A(I A) = (I A)A = 0; and, 5 A , (I A) and (I A ) are idempotent Example B1 Select a state estimation gain vector L such that the error state model for the discretetime system xk+1 pk = = 08187 00906 00000 00000 10000 10000 xk + xk 00906 00047 fk + L(pk pk )

