2. Matrix Methods

1. 2. Matrix Methods 2005. 3

2. Matrix • Definition of a Matrix • A set of numbers or other mathematical elements arranged in a rectangular array of rows and columns • A rectangular arrays of numbers arranged in m rows and n columns • Matrix usability • Solve complex systems of equations • Represent geometric objects in computer data bases • Perform geometric transformation • translation, rotation, scaling

3. Matrix • Matrix representation • denote a matrix with a boldface uppercase letters such as A, B, C, P, …, T (진한 대문자로 표시) • The elements of the matrix  lowercase subscripted letter • Ex) a32  the third rows and second column aij  row i and column j • May use a comma between subscript numbers

4. Matrix • Linear equation by a matrix Ex) x – 3y + z = 5 4x + y – 2z = -2 -2x + 3y = 1 AX = B A (3x3 Matrix) X (3x1) B (3x1)

5. Special Matrices • Square matrix • The number of rows equals the number of columns (m=n) • Row matrix • A single row of elements • Column matrix • A single column of elements • Diagonal matrix • A square matrix that has zero elements everywhere except on the main diagonal • Runs from the upper-left-corner to the lower-right-corner element • Scalar matrix • If all the aii are equal, then the diagonal matrix is a scalar matrix

6. Special Matrices • Identity matrix (단위행렬) • A special diagonal matrix that has unit elements on the main diagonal • Denote by the symbol I • Elements of I are denoted by • Kronecker delta • Null matrix • One whose elements are all zero • Symmetric matrix • A matrix whose elements are symmetric about the main diagonal • Antisymmetric matrix (= skew symmetric) • Transpose matrix (전치행렬) • Interchanging the rows and columns of a matrix (AT)

7. Matrix Equivalence & Arithmetic • Matrix Equivalence • Two matrix equal if all of their corresponding elements are equal • Matrix Arithmetic • Matrix addition • Commutative (교환법칙) • Scalar multiplication • Matrix multiplication • If and only if the 1st matrix is equal to the number of the rows of the 2nd matrix • Ex) A (m x n), B (n x p)  C (m x p) • Not commutative (교환법칙)

8. Matrix Equivalence & Arithmetic • Matrix addition & scalar multiplication • A + B = B + A • A + (B + C) = (A + B) + C • b(A + B) = bA + bB • (b + d)A = bA + dB • b(dA) = (bd)A = d(bA) • Matrix multiplication • (AB)C = A(BC) • A(B + C) = AC + AC • (A + B)C = AC + BC • A(kB) = k(AB) (kA)B • Matrix transpose • (A + B)T = AT + BT • (kA) T = kAT • (AB)T = BTAT • If AAT = I, then A is an orthogonal matrix(직교행렬)

9. Partitioning a Matrix • Partitioning a Matrix • Treat it as a matrix whose elements are these submatrices Ex) paritition T into the four submatrices T11, T12, T21, T22 • Eq) 2.39 – 2. 40 • Adding partitioned matrices • Eq) 2.41 – 2.42 • Multiplying partitioned matrices • Eq) 2.43 – 2.45

10. Determinants • Determinant (행렬식) • An operator in the form of a square array of numbers that produce a single value Ex) The determinant of a 2x2 matrix A  |A| Ex) The determinant of a 2x2 matrix A • Minor of an element aij of a determinant |A|  • Obtained by deleting elements of the i th row and j th column of |A| • Cofactor an element aij of a determinant |A|  cij • Obtained by the product of the minor of the element with a sign

11. Determinants • Properties of determinants • The value of a determinant is equal to the sum of the products of each element of any row (or column) and its cofactor • The determinant of a square matrix is equal to the determinant of its transpose: |A| =|AT| • Interchanging any two rows (or any two columns) of A change the sign of |A| • If we obtain B by multiplying one row (or column) of A by a constant, k, then |B| = k|A| • If two rows (or columns) of A are identical, then |A| = 0 • If we derive B from A by adding a multiple of one row (or column) of A to another row (or column) of A, then |B|=|A| • If A and B are both n x n matrices, then the determinant of their product is |AB| = |A||B| • If every element of a row (or column) is zero, then the value of the determinant is zero • If the determinant of a square matrix A is equal to one, • |A|=1, then it is orthogonal and proper • |A|=-1, then it is orthogonal and improper |A| > 0  proper matrix |A| < 0  improper matrix |A| = 0  degenerate matrix Nonsingular singular

12. Matrix Inversion • Matrix arithmetic • Matrix arithmetic does not define a division operation • But, include a process for finding the inverse of a matrix • The inverse of a square matrix A is A-1 • AA-1 = A-1A = I • The elements of A-1 are aij-1 Ex) for A-1 to exist at all, |A|!=0

13. Matrix Inversion • Using Matrix Inversion • Solving an algebraic equation • Eq) 2.57 – 2.58 • Matrix algebra using inversion • Eq) 2.59 – 2.66 • It would not work if |A| =0

14. Scalar and Vector Products • Use matrices to represent vectors • Scalar product • A=[a1 a2 a3], B=[b1 b2 b3] • Vector product Antisymmetric matrix

15. Transformed vector (n x n matrix) (n x 1 column matrix) eigenvector eigenvalue Eigenvalues and Eignevectors • Eigenvalues and Eignevectors • Eigenvector of A • Every vector (P) for which this is true for a given A • Eigenvalue of A • Lamda(l) is the eigenvalue of A corresponding to the vector P • Enginvalue (from the German eigenwerte  proper value)

16. Eigenvalues and Eignevectors • Characteristic equation • Which has nontrivial solution if P!=0 • Its solution are eigenvalues(li) of A • Ex) Eq. 2.75-2.84 (Characteristic equation) Characteristic equation eigenvalues Using the eigenvalue, can compute values of the corresponding eigenvectors Generalization  Ex) Eq. 2.85-2.88

17. Similarity Transformation • Similarity Transformation • a matrix is premultiplied and postmultiplied by another matrix and its inverse ( B = TAT-1 ) • A and B are similar matrices • Similar matrices • equal determinants • The same characteristic equation • The same eigenvalues, but not necessarily the same eigenvectors The eigenvalues of D are the eigenvalues of A l: eigenvalues of A S: nonsingular matrix Diagonal Matrix

18. Symmetric Transformation • Real symmetric matrix • aij =aji, or AT=A • If A and B are symmetric • [AB]T = BTAT = BA • If A is a real symmetric matrix • R-1AR is a diagonal matrix (R: orthogonal matrix)

19. Diagonalization of a Matrix • Diagonalization of a Matrix • E: square matrix of order n whose columns are the eigenvector Pi of A (nonsingular matrix) • L : a diagonal matrix whose elements are the eigenvalues of A Diagonalization of the matrix A