Vector Algebra

  • is a scalar. is an matrix.

Matrix can be

  • Symmetric

    • Positive-definite
    • Positive-semi-definitie
    • Negative-definite
    • Negative-semi-definite
  • Diagonal: entries outside the main diagonal are all zero

    • Diagonalizable: is diagonalizable iff exists invertible matrix where is a diagonal matrix
      • Inverse of a diagonalizable matrix is also diagonalizable
      • Orthogonally diagonalizable: is orthogonally diagonizable iff exists orthogonal matrix where
  • Invertible vs Singular

    • Invertible: There exists an inverse = determinant is non-zero
      • The inverse of a symmetric matrix is also symmetric
    • Singular: There does not exist an inverse
  • Orthogonal: is orthogonal iff

  • Rank:= dimension of the vector space generated by its columns

  • Identity Matrix

  • Determinant: scalar value that determines if the matrix has a determinant

Matrix Algebra Identities

  • Non-commutative
    • Commutative only with scalars
  • Distributive (w.r.t. matrix addition)
  • Associative
  • Transpose-distribution
  • Inverse-distribution
  • Transpose-Inverse