2. Trace and Norm Pertemuan 8 Matakuliah: MATRIX ALGEBRA FOR STATISTICS Tahun: 2009

3. Bina Nusantara University 3 Defenisi If A = (a i j ) is an n x n matrix, then the sum of the diagonal elements is called the trace of A (trace A). trace A = = trace A T Let A be mxn, and let A - be any weak inverse of A. Trace(A - A) = trace(AA - ) = rank A. =

4. Let α be a scalar and A and B be matrices Then a) tr(A T ) = trace(A), b) trace(α A) = α trace(A), c) trace(A + B) = trace(A) + trace(B), d) trace(AB) = trace(BA), e) tr(A T A) = 0 if and only if A = (0). Bina Nusantara University 4

5. If A is real and symmetric, then If A has real eigenvalues, then (traceA) 2 ≤ Rank A trace(A 2 ). If A is an n x n real matrix with real eigenvalues and exactly t of them are nonzero, then (traceA) 2 ≤ t trace(A 2 ) Bina Nusantara University 5

6. Let A be an n x n real matrix a) If A has real eigenvalues, then (traceA) 2 ≤ rank(A) trace(A 2 ). b) If A is symmetric, (trace A) 2 = rank(A) trace(A 2 ) if and only if there is a non-negative integer k such that A 2 = kA. c) If A is symmetric, then A 2 = A if and only if rank A = trace A = trace(A 2 ). d) trace(A T A) ≥ trace(A 2 ), with equality if and only if A is symmetric Bina Nusantara University 6

7. If A is an n x n real or complex matrix, then A can be written as A= XY-YX for some nxn matrices X and Y if and only if traceA = 0. Bina Nusantara University 7

8. Let A be m x n and B be n x m, both real or complex matrices. a) trace(AB) = trace(BA) = trace(A T B T ) = trace(B T A T ) b) If rn = n and either A or B is symmetric, then trace(AB) = Bina Nusantara University 8

9. Matix Then Bina Nusantara University 9

10. NORM Used for measuring distance in vector spaces and for providing a measure of how close one matrix is to another. Used for finding the best approximation of a matrix in a given class of matrices by a matrix in another class (e.g., of lower rank). Used for investigating limits of matrix sequences and series. Useful in statistics in the areas of inequalities, optimization, matrix approximation, matrix analysis, and numerical analysis. Bina Nusantara University 10

11. A function IIAII defined on all mxm matrices A, real or complex. is a matrix norm if the following conditions hold for all mxm matrices A and B. a) IIAII ≥0 b) IIAII = 0 if and only if A = (0). c) IIcAIl = IcIllAIl for any complex scalar c. d) IIA + BII ≤ IlAIl + IIBII e) IlABII ≤ IlAII IIBII Bina Nusantara University 11