This Week

1
This Week
2
Short Course in Mathematics and Analytic Geometry

• Week 6
• Matrices (Part 1)

3
Matrices

• In the context of matrices, elements are any
  abstract quantities that can be added and
  multiplied.
• A matrix is a rectangular table of elements.
• Each element has a row (i) and column (j) index.
• A matrix with m rows and n columns is called an
  m-by-n matrix, where m and n are its dimensions.

4
Scalar Product

• Given a scalar ? and a matrix A, the scalar
  product is given as
• For example

5
Matrix Addition

• Given two m-by-n matrices A and B, their sum is
  given by
• For example

6
Matrix Multiplication

• Multiplication of two matrices is only defined if
  the number of columns of the left matrix is equal
  to the number of rows of the right matrix.
• Given two matrices A and B, such that A has n
  columns and B has n rows, their product is given
  as

7
Matrix Multiplication

• For example

8
Properties of the Matrix (, ?)
9
Identity Matrices

• A square matrix is a matrix with dimensions of
  the form n-by-n.
• Given a matrix A, an identity matrix I is a
  square matrix such that AI IA A.
• For example

10
Matrix Transpose

• The transpose of a matrix A, written AT, is
  defined as follows
• For example

11
Special Transpose Matrices

• A matrix A is symmetric if A AT.
• A matrix A is called an orthogonal or a rotation
  matrix if AT is the inverse of A. That is, if
  AAT I.
• A matrix A is skew-symmetric if -A AT

12
Diagonal Matrices

• A square matrix A is diagonal if the only
  non-zero elements extend along the top-left to
  bottom-right diagonal.
• For example
• Diagonal matrices will be important for finding
  Eigen-values and Eigenvectors.

13
Determinant of a Matrix

• If one views a square matrix A as a set of row
  vectors, then the determinant of A, written
  det(A), is the volume of the parallelepiped,
  whose corner position vectors are these row
  vectors and their sums.

14
Minors of a Matrix Determinant

• A minor Mi,j is a reduced determinant found by
  omitting the ith row and jth column of a larger
  determinant. For example

15
The Cofactor of Determinants

• A cofactor Ci,j is a minor Mi,j augmented with a
  sign rule for the particular purpose of solving
  matrix determinants. Cofactors are defined as
  follows

16
Determinants by Minor Decompositions
17
Solving Simultaneous Equations

• Given a set of linear equations
• We can rewrite this as
• And is only solvable if

18
Cramers Rule

• Given that determinant D is not zero, the set of
  linear equations can be solved using Cramers
  rule

19
Vector as Matrices

• Matrices can completely represent vectors

20
Next Week

• Matrices (Part 2)
• Inverse matrices
• Eigen-values and Eigenvectors
• transformation Matrices