3 Matrices—Some Definitions A matrix is a rectangular array of real numbers.Denote a matrix with an upper case italic letter, e.g.,
4 Matrices—Some Definitions A has m rows and n columns.The dimension of A is “m x n” (m by n).A vector is a matrix with a single row or a single column (denoted with a lower case letter without a subscript), e.g,
5 Matrices—Some Definitions If n = m then A is a square matrix.If A is square and aij = aji then A is symmetric, e.g.,is a symmetric matrix.
6 Scalar Multiplication Let k be a scalar (i.e., a given real number). kA is a new matrix where each element equals kaij.
7 Matrix AdditionA + B is a new matrix where each element is the sum of the corresponding elements of A and B.
8 Matrix AdditionAddition is only defined if the matrices have the same dimensions.makes no sense.
9 Matrix SubtractionA – B = A + (–1)B. So subtraction also requires the matrices to have the same dimensions.
10 Matrix TransposeThe transpose of a matrix is a new matrix where the rows and columns are switched. The transpose of A is denoted A .
11 Matrix Rules for Addition A + B = B + A(A + B) + C = A + (B + C)(A + B) = A + B
12 Summation OperatorWe often need to add up the elements of a vector or matrix. We use a convenient notation for this.Suppose a = (a1, a2, …, an). Define
13 “Dot” (or “Inner”) Product The “dot” product “multiplies” two vectors (we ignore other vector multiplication concepts).The dot product of a and b is defined as ab = ∑aibi (a and b must have the same number of elements).Supposethen ab = 1 x x 2 + 2(-3) = 2.
14 Matrix Multiplication Matrix multiplication AB is defined in terms of dot products. The result is a new matrix C.Each cij is a dot product.The dot product involves the ith row of A and the jth column of B.
22 Singular Matrix For = 0, –1 does not exist. Sometimes A –1 does not exist. In this case A is called singular or non-invertible.Excel sometimes calculates an inverse of a singular matrix when there is a lot of roundoff error—be aware!