Basic Matrix Operations

• Addition

  
  
  
  

  Can only add a matrix with matching dimensions, or a scalar.

• Scaling

  
  

• Dot product of vectors

  
  
  
  

  Inner product (dot product) x·y is also |x||y|Cos( the angle between x and y )

• Multiplication

  
  
  

  By convention, we can refer to the matrix product AA as A2, and AAA as A3, etc.
  Obviously only square matrices can be multiplied that way

• Transpose

  
  
  
  A useful identity:

• Inverse / pseudoinverse
  Given a matrix A, its inverse A^-1 is a matrix such that AA^-1 = A^-1A = I
  E.g.
  

  
  Inverse does not always exist. If A^-1 exists, A is invertible or non-singular. Otherwise, it’s singular.
  Useful identities, for matrices that are invertible:
  
  Pseudoinverse
  Say you have the matrix equation AX=B, where A and B are known, and you want to solve for X
  You could use MATLAB to calculate the inverse and premultiply by it: A^-1AX=A^-1B → X=A^-1B
  MATLAB command would be inv(A)*B
  But calculating the inverse for large matrices often brings problems with computer floating-point resolution (because it involves working with very small and very large numbers together).
  Or, your matrix might not even have an inverse.
  Fortunately, there are workarounds to solve AX=B in these situations. And MATLAB can do them!
  Instead of taking an inverse, directly ask MATLAB to solve for X in AX=B, by typing A\B
  MATLAB will try several appropriate numerical methods (including the pseudoinverse if the inverse doesn’t exist)
  MATLAB will return the value of X which solves the equation:
  If there is no exact solution, it will return the closest one
  If there are many solutions, it will return the smallest one
  MATLAB example:
  
  
  

• Determinant / trace
  det(A) returns a scalar
  Represents area (or volume) of the parallelogram described by the vectors in the rows of the matrix

  
  
  For
  
  Properties:
  Trace
  
  
  

  Invariant to a lot of transformations, so it’s used sometimes in proofs. (Rarely in this class though.)
  

  Properties:

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2D Transformation

Basic set of 2D planar transformations.

Homogeneous system

In general, a matrix multiplication lets us linearly combine components of a vector

• This is sufficient for scale, rotate, skew transformations.

• But notice, we can’t add a constant! :(

  
  Hierarchy of 2D coordinate transformations.

• translation

  
  

• rigid(Euclidean) = rotation + translation

  
  
  

  Note: R belongs to the category of normal matrices and satisfies many interesting properties:

  
  

• similarity transform = scaled rotation

  
  

• affine

  
  
  

  Parallel lines remain parallel under affine transformations.

• projection

  
  
  

  Perspective transformations preserve straight lines

3D Transformation

Hierarchy of 3D coordinate transformations

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Singular Value Decomposition (SVD)