Matrices and Their Operations

from Tokyo University 1S1 Mathematical Foundations: Linear Algebra

Matrices and their operations

• 
• Sometimes written as $A=(a_{i,j})$
• Terminology
  • Square matrix
    • A matrix with the same number of rows and columns
    • Diagonal components: Elements on the diagonal line from top left to bottom right
    • Diagonal matrix: A square matrix where all elements except the diagonal components are 0
      • We can say that an n-dimensional diagonal matrix and an n-dimensional vector are in a one-to-one correspondence relationship (blu3mo)
  • Identity matrix
    • In Japanese, it is a diagonal matrix with all diagonal components being 1
    • 
    • Corresponds to the number 1 in real numbers and has the property of “not changing anything when multiplied”
    • Kronecker’s delta
      • 
      • That symbol
        • ${\delta }_{i,j}$ is equal to the $(i,j)$ component of the identity matrix
  • Zero matrix
    • Corresponds to the number 0 in real numbers and has the properties of “not changing anything when added” and “becoming 0 when multiplied”
  • Operations
    • Scalar multiplication and addition have definitions that are quite intuitive (blu3mo)
    • Matrix multiplication
      • Not all matrices can be multiplied together
      • The easiest and most intuitive way is to draw a horizontal line above the first matrix and a vertical line to the left of the second matrix
      • It can be defined between an l x m matrix and an m x n matrix
        • The result will be an l x n matrix
        • I see~ (blu3mo)
        • Each block must be a vector of the same dimension when separating with a line
      • Does this have anything to do with the cross product of vectors?
        • But with this definition, multiplication between vectors is not possible, right?
        • Then is this a different operation from that?
      • Points to note
      • Rules
        • Associative law
          • $(AB)C=A(BC)$
        • Distributive law
          • $A(B+C)=AB+AC$ and $(A+B)C=AC+BC$ hold
          • (Since the commutative law does not hold, it is necessary to distinguish between the two)
        • ⭐️Commutative law does not hold
          • $AB\ne BA$
          • Various other operation rules hold subtly, so this is actually a trap (blu3mo)
          • Well, it seems like we only need to be concerned about this
        • ⭐️But, this holds:
          • $(\lambda A)(\mu B)=(\lambda \mu )(AB)$ ($\lambda$ is a scalar)
  • Impressions
    • It’s interesting how all the definitions can be derived from wanting to represent the multiplication of matrices and vectors as Linear transformations on a plane
      • It’s amazing how everything comes together so coherently
    • Different uses for notes
      • Tricky problems can be done in GoodNotes, which I often used for notes in IB
      • Concept explanations can be done in Scrapbox
• Applications of matrices
  • Matrices and Linear Transformations
  • Coefficient matrix of a system of linear equations
    • We can express a system of simultaneous linear equations as $A\vec{x}=\vec{b}$