Linear Algebra - Matrix Matrix (Multiplication)

Vector-matrix Linear Combination

A Linear combinations definition of vector-matrix multiplication (Ie the A vector is seen as the coefficient container that must be applied to the others vectors) <MATH>\alpha_1.[b_1] + \alpha_2.[b_2] + \alpha_3.[b_3]</MATH>

Implementation Pseudo-Code:

# Transform the matrix as Row Vectors
rowVectorDict = mat2rowdict(M)
# Multiply the row vector by the coefficient of the corresponding vector
rowVectorsWithCoef = [v[j]*rowVectorDict[j] for position j in rowVectorDict of B]
# Addition the vector
resultVector = sum(rowVectorsWithCoef)

Total Example:

def linear_combination_vector_matrix_multiplication(v, M):
    assert(v.D == M.D[0])
    rowVectorDict = mat2rowdict(M)
    rowDictTimesVec = [(v.f[iDict] if iDict in v.f else 0)*rowVectorDict[iDict] for iDict in rowVectorDict]
    return sum(rowDictTimesVec)

Vector-matrix Dot product

A Dot-product definition of vector-matrix multiplication is the multiplication of two vectors.

<MATH> \begin{array}{c} [1, 0, 0] * \begin{bmatrix} b_1 \\ \hline b_2 \\ \hline b_3 \end{bmatrix} = b_1 {\bf \text{ and } } [2, 1, 0] * \begin{bmatrix} b_1 \\ \hline b_2 \\ \hline b_3 \end{bmatrix} = 2b_1+b_2 {\bf \text{ and } } [0, 0, 0] * \begin{bmatrix} b_1 \\ \hline b_2 \\ \hline b_3 \end{bmatrix} = b_3 \\ {\bf \text{ therefore }} \begin{bmatrix} 1 & 0 & 0 \\ \hline 2 & 1 & 0 \\ \hline 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} b_1 \\ \hline b_2 \\ \hline b_3 \end{bmatrix} = \begin{bmatrix} b_1 \\ \hline 2 b_1 + b_2 \\ \hline b_3 \end{bmatrix} \end{array} </MATH>

<math> \begin{bmatrix} 1 & 0 & 0 \\ \hline 2 & 1 & 0 \\ \hline 0 & 0 & 1 \end{bmatrix} </math> is called an elementary row-addition matrix.

By matrix-vector definition of matrix-matrix multiplication, the result is a matrix with one column that you can interpret as a vector : a column vector

Implementation Dot Product Pseudo-code:

# Get the columns vector
# Dot product multiplication with the vector

Implementation:

def dot_product_vector_matrix_multiplication(v, M):
    assert(v.D == M.D[0])
    columnVectors = mat2coldict(M)
    return Vec(M.D[1],{i:columnVectors[i]*v for i in M.D[1]})

Matrix-vector Linear Combination

Pseudo-Code:

• Transform the A matrix as Column Vectors

<math> \begin{array}{rrr} columnVectorA_1 & = & \begin{bmatrix}1\\-1\end{bmatrix}\\ columnVectorA_2 & = & \begin{bmatrix}2\\1\end{bmatrix} \end{array} </math>

• Transform the B matrix as Column Vectors

<math> \begin{array}{rrr} columnVectorB_1 & = & \begin{bmatrix}4\\3\end{bmatrix}\\ columnVectorB_2 & = & \begin{bmatrix}2\\1\end{bmatrix} \\ columnVectorB_3 & = & \begin{bmatrix}0\\-1\end{bmatrix} \end{array} </math>

• Multiply each A column vector by the coefficient of the corresponding column vector of B to make a linear combination and addition the vector. Example for the first column vector of B (ie B1):

<math> \begin{array}{rrllll} columnVectorAB_1 & = & columnVectorB_1[0] * columnVectorA_1 & + & columnVectorB_1[1] * columnVectorA_2 \\ & = & 4 * columnVectorA_1 & + & 3 * columnVectorA_2 \\ & = & 4 * \begin{bmatrix}1\\-1\end{bmatrix} & + & 3 * \begin{bmatrix}2\\1\end{bmatrix} \\ & = & \begin{bmatrix}4*1\\4*-1\end{bmatrix} & + & \begin{bmatrix}3*2\\3*1\end{bmatrix} \\ & = & \begin{bmatrix}4\\-4\end{bmatrix} & + & \begin{bmatrix}6\\3\end{bmatrix} \\ & = & \begin{bmatrix}10\\-1\end{bmatrix} \end{array} </math>

• and restart the process for the next column vector of B to get the full matrix:

Total Example:

def linear_combination_matrix_vector_multiplication(M, v):
    # assert the column domain is the domain of the vector
    assert(M.D[1] == v.D)
    colDict = mat2coldict(M)
    colDictTimesVec = [(v.f[iDict] if iDict in v.f else 0)*colDict[iDict] for iDict in colDict]
    return sum(colDictTimesVec)

Matrix-vector Dot Product

<MATH>\text{AB is the matrix with:}</MATH> <MATH>\text{column i of } AB = A * ( \text{column i of } B)</MATH>

<MATH> \begin{array}{lllllllllll} AB_{J1} & = & A * [4,3] & = & [[1,2]*[4,3], [-1,1]*[4,3]] & = & [1*4+2*3, -1*4+1*3] & = & [10,-1] \\ AB_{J2} & = & A * [2,1] & = & [[1,2]*[2,1], [-1,1]*[2,1]] & = & [1*2+2*1, -1*2+1*1] & = & [4,1] \\ AB_{J3} & = & A * [0,-1] & = & [[1,2]*[0,-1], [-1,1]*[0,-1]] & = & [1*0+2*-1, -1*0+1*-1] & = & [-2,-1] \end{array} </MATH> <MATH>AB = \begin{bmatrix} \begin{array}{r|r|r} 10 & 4 & -2 \\ -1 & 1 & -1 \end{array} \end{bmatrix} </MATH>

Implementation for one vector matrix:

def dot_product_mat_vec_mult(M, v):
    assert (M.D[1] == v.D)
    # Get the row vectors
    rowVectors = mat2rowdict(M)
    # Dot product multiplication with the vector
    return Vec(M.D[0],{i:rowVectors[i]*v for i in M.D[0]})