Matrix Algebra

Linear algebra review, part 2: matrices as collections of vectors (builds on Linear Vector Spaces).

Matrices and Partitions

• Matrix , :
  • , row index , column index

Column Partition

• Each column is a vector :
• This is the column partition of

Row Partition

• Each row is a (transposed) vector , with (rows of ):

Matrix-Vector Multiplication

• Given , , the product
• Defined only if the size of () = # of columns in ()
• If so, then ( = # rows in ), and is a linear combination of the columns of
• Equivalently, using the row partition:
  • an array of dot products with the rows of
• The set of all such is the range of

Matrix-Matrix Multiplication

• Given , , the product
  • is defined only if (the inner dimensions must match), and then
  • has size (the outer dimensions)
• Equivalently
  • (dot product of row of with column of )

Square and Identity Matrices

• “Square” matrix: the matrix has the same # of rows as columns
• “Identity matrix”: ,
  • for any ,
  • for any ,

Matrix Inverse

• For , matrix is an inverse if
• If such an exists we denote it by
• Computationally:
  • is a scalar; is an matrix of sub-determinants
  • Both can be recursively reduced to the case:
• For

Possible slip in notes

• The handwritten determinant reads ; the second term should be (written correctly above)

NOTE

• is undefined if . Such a matrix is singular
• See Singular Matrices for equivalent characterizations