Matrix

Operation

  • Addition and Subtraction

    Only matrices with same dimension could be added/substract. Doing the add/subtract means each number in the first matrix will add/subtract the number in the same position in the second matrix.

  • Scalar Multiplication

    A real number multiply a matrix means each of the number in the matrix will be multiplied with that real number.

  • Matrix Vector Multiplication

    Let say we are going to mulitply a 3x2 matrix A with an 2x1 vector x, the result is y which is a 3x1 vector. So to get y[i], multiply A's ith row with elements of vector x, and add them up.

  • Matrix Matrix Multiplication

    Let say we are going to mulitply a mn matrix *A with an no matrix *B, the result is C which is a mo matrix. So the *ith column of the C is obtained by multiplying A with the ith column of B. (for i=1,2,…,o)

    In short, Matrix and Matrix multiplication result is the combination of the result from Matrix with several Vectors.

  • Matrix Multiplication

    No commutative law for Matrix multiplication. A x B not equal to B x A

    Has association law for Matrix multiplication. (A x B) x C equal to A x (B x C)

  • Identity Matrix

    Always mark Identity matrix as I. I x A = A x I = A

    The size of the I is implict according to the context. Let say A is a 3x2 matrix, than A x I, the I has a 2x2 size. I x A, the I has a 3x3 size.

  • Matrix Inverse

    If A is an m x m matrix, and if it has an inverse, than A x A(-1) = A(-1) x A = I.

  • Matrix Transpose

    Let A be an m x n martrix, and let B = A(T). Then B is an n x m matrix, and B(ij) = A(ji).

Vector

Vector is a matrix with only on column.