1. 2. Matrix Methods

2. Matrix Definition of a Matrix Matrix usability
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
AX = B A
(3x3 Matrix) X
(3x1) B
(3x1)

5. Special Matrices Square matrix Row matrix Column 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) Null matrix Symmetric matrix
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
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. for A-1 to exist at all, |A|!=0
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
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. 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)
Transformed vector
(n x 1 column matrix)
(n x n matrix)
eigenvector
eigenvalue

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

17. 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
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