Matrix operations

University:
University of Michigan
Course:
MATH 217 | Linear Algebra
Academic year:

2022
Views:

327

Pages:

16
Author:

Anderson Carpenter

Matrix operations Throughout this course, we’ll be doing computations with matrices. We’ll be talking a lot about why matrices work the way they do, but for right now, let’s learn how they work. A matrix is a rectangular grid of numbers.   7 5 −1 12 0   3 4 −2 0 0   0 −9 3 10 0 A row of a matrix is a horizontal band of entries; a column is a vertical band. 5 −1 12 4 −2 0  0 ←− Row 1  0 ←− Row 2 −9 3 10 0  7  3 0 ←− Row 3 Column 1 Column 2 Column 3 Column 4 Column 5 ↓ ↓ ↓ ↓ ↓ 7 5 −1 12 0    3 4 −2 0 0   −9 3 10 0  0 We call a matrix m × n where m is the number of rows and n is the number of columns. This is a 3 × 5 matrix:  7  3  0 5 −1 4 −2 −9 3 We label the entries of a matrix A as and j is the column label:  A11 A12 A13  A21 A22 A23  A31 A32 A33  12 0  0 0  10 0 Aij where i is the row label  A14 A15 A24  A25   A35 A34 We can only add two matrices when they have the same size. In that case, we just add their entries:     B B12 B13 A A12 A13 =  11  +  11 B21 B22 B23 A21 A22 A23   A + B11 A12 + B12 A13 + B13  11 . A21 + B21 A22 + B22 A23 + B23 We multiply two matrices when the number of columns of the first matrix equals the number of rows of the second matrix. We multiply two matrices when the number of columns of the first matrix equals the number of rows of the second matrix. I like to think of a matrix as a machine having output pipes hanging off the lefthand side, and input pipes coming in from the right through the top:  A11  A21  A31 A12  A13 A14 A22 A23 A32 A33  A24   A34 We multiply two matrices when the number of columns of the first matrix equals the number of rows of the second matrix. We can multiply two matrices when the number of input pipes of the first matrix matches the number of outputs of the second:  A11  A21  A31 A12  A13 A14 A22 A23 A32 A33  A24   A34  B11  B  21  B31  B41 B12   B22    B32   B42 We multiply two matrices when the number of columns of the first matrix equals the number of rows of the second matrix. We can multiply two matrices when the number of input pipes of the first matrix matches the number of outputs of the second: The product of an ` × m matrix by a m × n matrix is an ` × n matrix. In other words, jam together the pipes. A11 A12 A13 A14 A21 A22 A23 A24 A31 A32 A33 A34 × " B11 B21 B31 B41 B12 B22 B32 B42 # = C11 C12 C21 C22 C31 C32 To compute entry (i, k) of the product, add up the products going across the i-th row of the first matrix and down the k-th column of the second matrix: A11 A12 A13 A14 A21 A22 A23 A24 A31 A32 A33 A34 " × B11 B21 B31 B41 B12 B22 B32 B42 # = C11 C12 C21 C22 C31 C32 C21 = A21 B11 + A22 B21 + A23 B31 + A24 B41 . If you like letters better than numbers, X Cik = Aij Bjk . j Why? See the next lecture, and basically the whole course! It is important to note that it matters what order we multiply matrices in! If AB is defined, the product BA need not even be defined: h∗i h∗i ∗∗∗ ∗ ∗∗ ∗ [∗ [∗ ∗ ∗] ∗ = [∗] ∗ ∗ ∗ ] doesn’t make sense! ∗ ∗ Even if AB and BA are defined, they needn’t have the same size: [ ∗∗ ] [ ∗ ∗ ] = [ ∗∗ ∗∗ ] [ ∗ ∗ ] [ ∗∗ ] = [ ∗ ] . And, even if AB and BA have the same size, they needn’t be equal: h i h i a b w x w x a b aw+by ax+bz aw+cx bw+dx [ y z ] c d = ay+cz by+dz . c d [ y z ] = cw+dy cx+dz Properties of matrix operations I told that, with matrices, A × B need not be the same thing as B × A. In my experience, most students get used to keeping track of this issue while being comfortable that matrix addition and multiplication “mostly” work the way you expect. But a few students want a detailed list of exactly which things still work. The point of this lecture is to give you that list. If you like the idea of studying which properties hold and do not hold for a new type of mathematical object, and the consequences of that, you want to study Abstract Algebra. Take Math 312 or 412. Properties of matrix operations Whenever the following equations make sense, they are true: A+0=0+A=A Id X = X X Id = X A0 = 0 0A = 0 A+B =B+A (A + B) + C = A + (B + C) (AB)C = A(BC) A(B + C) = AB + AC (X + Y )Z = XZ + Y Z where ··· 0  0 0 0 ··· 0  0 0 0 ··· 0  0 0 0 0= .. . 0 0 0 ··· 0 ··· 0  0 1 0 ··· 0  0 0 1 ··· 0  1 0 0 and Id = .. . 0 0 0 ··· 1 Transpose The transpose of a matrix is that matrix with the rows and columns switched. We write AT for the transpose of A. For example,    T a d   a b c    = b e   d e f c f We have (AB)T = B T AT . Transpose will really come into its own in Chapter 5, but the book introduces it now, so we will too. Also, there is one very small thing it is useful for immediately . . . Transpose We (and your textbook, and most of the world) will generally write vectors as column matrices: So a three dimensional vector would be hxi y . We usually put an arrow over vectors, like this ~ v = [ vv12 ]. z This makes most formulas more convenient, but makes one thing less convenient: Vectors with lots of entries take up a lot of space on the page:  x1  x2  xx34  x5  x6  x7 x8 x9 x10   .  We can use transpose to solve this notational issue, and write [ x1 instead. x2 x3 x4 x5 x6 x7 x8 x9 x10 ] T

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