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Matrix

  • A matrix is rectangular array of numbers
  • 𝑚 rows by 𝑛 columns → 𝑚×𝑛 matrix
    • 𝐴=[142536]2×3

Reading a Matrix

  • 𝐵=[387513]
  • Elements are identified as 𝐵row,column
    • 𝐵2,25

Matrix Laws

Adding/Subtracting Matrices

  • Must have the same dimensions
  • [1324]+[5768]=[1+53+72+64+8][610812]

Dot Product of Vectors

  • The sum of the element-wise multiplication
  • [1234][[[5678]]]=1×5+2×6+3×7+4×870

Multiplying a Matrix and Vector

  • Numbers of columns in the matrix must be the number of rows in the vector
  • [142536]×[789]=[1×7+2×8+3×94×7+5×8+6×9][50122]

Multiplying Two Matrices

  • Rows of 𝐴 must equal columns of 𝐵
  • (𝐴×𝐵)𝑖,𝑗=𝑛𝑘=1𝐴𝑖,𝑘×𝐵𝑘,𝑗
  • [142536]×[791181012]=[1×7+2×9+3×114×7+5×9+6×111×8+2×10+3×124×8+5×10+6×12][5813964154]

Identity Matrix

  • Identity matrix 𝐼 acts as 1 in multiplication
  • Anything times 𝐼 is itself

Transposition/Rotating

  • Swap rows and columns
  • [135246]𝑇=[123456]

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