from Tokyo University 1S1 Mathematical Foundations: Linear Algebra

Matrices and their operations

  • image
  • Sometimes written as
  • 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
      • image
      • Corresponds to the number 1 in real numbers and has the property of “not changing anything when multiplied”
      • Kronecker’s delta
        • image
        • That symbol
          • is equal to the 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
          • Distributive law
            • and hold
            • (Since the commutative law does not hold, it is necessary to distinguish between the two)
          • ⭐️Commutative law does not hold
            • 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:
            • ( 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