Week 3 - Matrix-Vector Operations

1. Special Matrices

  • Special Vectors:

    • Unit Vector: Any vector of length one (unit length). For example, the vector (2222)\begin{pmatrix}\frac{\sqrt{2}}{2} \\ \frac{\sqrt{2}}{2}\end{pmatrix} has length one.
    • Standard Unit Vector:

2. Triangular Matrices

3. Transpose Matrix

4. Symmetric Matrix

5. Scaling a Matrix

6. Adding Matrices

7. Matrix-vector Multiplication

8. Cost of Matrix-Vector Multiplication

  • Consider y:=Ax+y , where ARm×ny := Ax+y\ \text{, where } A \in R^{m \times n} :
    • Notice that there is a multiply and an add for every element of A.
    • Since A has m×n=mnm \times n = mn elements, y:=Ax+yy := Ax+y, requires mn multiplies and mn adds, for a total of 2mn floating point operations (flops).

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