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COMMATH

COMMATH. Lecture 01 – Matrices Reggie C. Gustilo, MS EE. Outline. Review of Systems of Equations Matrix Definition and Notation Matrix Operations Determinants Inverse of a Matrix Eigenvalues and Eigenvectors. Review of Systems of Equations. Recall solving:.

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COMMATH

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  1. COMMATH Lecture 01 – Matrices Reggie C. Gustilo, MS EE

  2. Outline • Review of Systems of Equations • Matrix Definition and Notation • Matrix Operations • Determinants • Inverse of a Matrix • Eigenvalues and Eigenvectors

  3. Review of Systems of Equations • Recall solving:

  4. Review of Systems of Equations • How do we solve such systems of linear equations? • When do we use each method? • What is(are) the difficulty(ies) in solving such kind of problems? • Are there any alternatives in solving such problems?

  5. Matrix Notation

  6. Matrix Notation • Coefficient Matrix – contains the coefficients of the variables • Variable Matrix – contains the variables used in the system of equations • Constants – either zero or any real number

  7. Matrix Notation An m x n matrix m – number of rows n – number of columns aij – element in row I and column j

  8. Matrix Notation • When: • m = n  square matrix • Equal number of equations and variables • m > n  skinny matrix • Number of equations is greater than the number of variables • m < n  fat matrix • Number of equations is less than the number of variables

  9. Examples • Put the following information into a 3 by 2 matrix and attach labels: The Lions won 5 games and lost 8. The Tigers won 9 and lost 4. The Bears won 7 and lost 6. • Each team played 15 games. They either won, lost, or tied each game. Put the following information into a 3 by 4 matrix and attach labels: The Snakes won 6 and lost 8. The Lizards won 8 and lost 7. The Frogs won 9 and tied 2. The Toads lost 9 and tied 1.

  10. Matrix Operations • Scalar Multiplication • B = a*A • a = scalar number • A = given matrix • B = resulting matrix • A can be of any size. Size of B will follow the size of A.

  11. Example

  12. Matrix Operations • Addition and Subtraction • Recall that: • A = B ± C = C ± B iff size of B and C are the same, that is the number of rows and columns of B and C are the same. • Commutative property applies.

  13. Example

  14. Matrix Operations • Multiplication • Recall that: • A = B*C iff column of A is equal to the row of B. • If B = n x m and C = m x p, then A = n x p • Commutative property does not always apply.

  15. Example

  16. Matrix Operations • Transpose of a matrix • The row values become the column values and the column values become the row values. • AT = transpose of A • If A = nxm, then the transpose of A is then A = mxn

  17. Example

  18. Matrix Terminologies • Two matrices, A and B are equal if their corresponding elements are equal. • A matrix is said to be symmetric if AT = A. • What kind of matrix does this property hold? • An identity matrix is of the form:

  19. Matrix Terminologies • A lower triangular matrix is of the form: • An upper triangular matrix is of the form:

  20. Exercises • Perform what is asked. A + C D + E B – (A+C) A + DT AD - BE FT – C (D + E)T – (A + B) CF - DTAT

  21. Determinants • Limit to 2x2 and 3x3 matrices

  22. Determinants • When • Important when determining the inverse of a matrix. (later discussions)

  23. Properties of Determinants • The determinant of a matrix will be zero if • An entire row is zero. • Two rows or columns are equal. • A row or column is a constant multiple of another row or column. • If a square matrix B is obtained from a square matrix A by interchanging two rows (or two columns), then |B| = – |A|

  24. Properties of Determinants • If square matrix B is obtained from square matrix A by multiplying each element of any row (or column) of A by some real number k, then |B| = k|A|. • If square matrix B is obtained from square matrix A by adding k times a row (or column) of A to another row (or column) of A, then |B| = |A|.

  25. Determinants • Given the triangle, find its area.

  26. Example • The area is:

  27. Determinants

  28. Cofactor of a Matrix • For a square matrix A = [aij], the cofactor Aij of an element aij is given by: Aij = (-1)i+jMij • Where Mij is the minor of aij.

  29. Cofactor of a Matrix • Best understood through an example.  http://www.mathwords.com/c/cofactor_matrix.htm

  30. Adjoint of a Matrix • The matrix formed by taking the transpose of the cofactor matrix of a given original matrix. • The adjoint of matrix A is often written adj A.

  31. Example

  32. Inverse of a Matrix • There is no such division in matrix operations.  • For a square matrix A, the inverse is written A-1. When A is multiplied by A-1 the result is the identity matrix I. • A*A-1 = I • Non-square matrices do not have inverses. • Not all square matrices have inverses. • A square matrix which has an inverse is called invertible or nonsingular. • A square matrix without an inverse is called noninvertible or singular.

  33. Limit to 2x2 and 3x3 matrices Adjoint method Augmented matrix method Inverse of a Matrix

  34. Inverse of a Matrix • Augmented matrix method process • Performs elementary row operations

  35. Matrix Applications • Let’s try solving these using matrices.

  36. Matrix Applications Are our answers here the same with our previous answers? 

  37. Eigenvalues and Eigenvectors • We recall that matrices represent real-life problems expressed as equations of the form: Ax = x • A – a square matrix of coefficients • x – a column matrix of state variables •  – a scalar quantity

  38. Eigenvalues and Eigenvectors

  39. Eigenvalues and Eigenvectors •  – eigenvalue, characteristic value or latent root of matrix A • Corresponding solution fo the given Ax = x are called eigenvectors or characteristic vectors of A

  40. Example 1 • Find the eigenvalues and corresponding eigenvectors when:

  41. Example 1 • Determine the eigenvalues.

  42. Example 1 • Determine the eigenvectors. • Substitute each eigenvalue to the equation (A-)x=0.

  43. Example 1 • Determine the eigenvectors. • Substitute each eigenvalue to the equation (A-)x=0.

  44. Example 2 • Find the eigenvalues and corresponding eigenvectors when:

  45. Example 2 • Let’s prove that the answers are as given below.

  46. In the real world… • Matrices have dimensions more than 3x3. • So, how do we find the following: • Inverse • Determinant • Transpose • Eigenvalues and Eigenvectors • Matrices do not only have real numbers as its elements.

  47. In the real world… • We have softwares such as: • Excel • Matlab/Octave/Scilab • Mathcad • Mathematica • Etc. • What is left is how do we analyze the results? 

  48. In the real world… • Its uses… • Control systems  state-space analysis • Communications  signal representations • Industrial Engineering  Linear programming • Graph Theory • Electronics  solution of unknown voltages and currents • And many more… 

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