Matrix and Vector Operations

Introduction to Matrices

• Definition: A matrix is an array of numbers, specifically a two-dimensional array.
• Dimensions: An m by n matrix has:
  • m: Number of rows
  • n: Number of columns

Structure of a Matrix

• General form of an m x n matrix A:
  • A = [ a₁₁ a₁₂ ... a_1n
    a₂₁ a₂₂ ... a_2n
    ...
    a_m1 a_m2 ... a_mn ]
• Total Entries: The total number of entries in the matrix = m * n*

Matrix-Vector Multiplication

• The operation we will focus on is multiplying a matrix A by a vector x.

Conditions for Multiplication

• The vector x must have the same number of components (n) as the number of columns in matrix A.
• Vector Form: x = [ x₁, x₂, ..., x_n ]

Definition of Matrix-Vector Product

• The product A * x results in a new vector b:
  • b₁ = a₁₁x₁ + a₁₂x₂ + ... + a_1nx_n
  • b₂ = a₂₁x₁ + a₂₂x₂ + ... + a_2nx_n
  • ...
  • b_m = a_m1x₁ + a_m2x₂ + ... + a_mnx_n*

Resulting Vector

• The resulting vector b has m entries, one for each row of the matrix A.

Dot Product Relation

• The product can be viewed as the dot product of row vectors of the matrix A with the vector x.
• This can be formally viewed using transposes:
  • A = [ a₁ a₂ ... a_m ]
  • a_i is the i-th row vector of A.
  • The operation A * x = a₁ • x, a₂ • x, ..., a_m • x*

Example Calculation

• Given a matrix A and vector x:
  • A = [ [-3, 0, 3, 2], [1, 7, -1, 9] ]
  • x = [2, -3, 4, -1]
• Calculation of A * x:
  • b₁ = -6 + 0 + 12 - 2 = 4
  • b₂ = 2 - 21 - 4 - 9 = -32
  • Result: b = [4, -32]*

Interpretation of Matrix-Vector Products

• Two interpretations:
  1. Dot Product Interpretation: Each entry of the resulting vector is a dot product of a row vector from A with vector x.
  2. Linear Combination Interpretation: The product A * x can be viewed as a linear combination of the column vectors of A, weighted by the components of vector x.*

Conclusion

• Understanding matrix-vector multiplication provides insight into how matrices can transform vectors and the underlying operations involved.