Understanding Matrix Operations and Notation in Mathematics
Explore the world of matrices in mathematics with this comprehensive guide covering matrix operations, notation, inner product, matrix multiplication, outer product, and transpose. Learn about the dimensions, sizes, and mathematical operations involved in matrix manipulation.
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A representational view of matrices Operation Operation D a t a D a t a D a t a D a t a Operation Operation Operation Output Operations Input
A note on notation m = number of rows, n = number of columns Number : Number of vectors Size : Dimensionality of each vector (elements) [Operation matrix] [Input matrix] [number of operations x operation size] [size of input x number of inputs] [m1 x n1] [m2 x n2] = [m1 x n2] [2 x 3] [3 x 2] = [2 x 2] n1 and m2 (where the matrices touch) have to match size of operations has to match size of inputs. The resulting matrix will have dimensionality number of operations x number of inputs.
Inner Product = Dot product Dot product = Row Vector * Column Vector = Scalar scalar DP = (m=n) [1xn] [mx1] [1x1] 5 6 7 8 1 2 3 4 1*5 + 2*6 + 3*7 + 4*8 70
Matrix multiplication 1 2 3 4 5 6 7 8 19 22 43 50 1 2 3 4 5 6 7 8 (1*5 + 2*7) (3*5 + 4*7) (1*6 + 2*8) (3*6 + 4*8) 3 4 3 4 1 2 1 2
The outer product What if we do matrix multiplication, but when the two matrices are a single column and row vector? OP = [mx1] [1xn] [mxn] Output is a *matrix*, not a scalar. 1 2 (1*3) (1*4) (2*3) (2*4) 3 4 6 8 3 4
The transpose = flipping the matrix 1 4 7 1 2 3 2 5 8 4 5 6 3 6 9 7 8 9 NOT just a rotation!
