Lecture 2 Matrix Operations
• matrix multiplication matrix-vector product. • matrix inverse. 2–1. Page 2 Matrix Operations. 2–5. Page 6. Matrix multiplication if A is m × p and B is p ...
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Here we will learn some basic matrix operations: Adding and Subtracting Transpose
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Addition: if A and B are matrices of the same size m × n then A + B
Array operations and Linear equations
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Basic Matrix Operations
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1.10 Mathematical Operations with Arrays 1.10.1 Array Operations
Matrix operations follow the rules of linear algebra. By contrast array operations execute element by element operations and support multidimensional arrays.
Lecture 2 Matrix Operations
matrix multiplication matrix-vector product. • matrix inverse. 2–1 we can multiply a number (a.k.a. scalar) by a matrix by multiplying every.
Appendix A: Summary of Vector/Matrix Operations
?. ?. ?. ?. ?. 18. 2. 42 1 . Matrix multiplication is not commutative; that is C = AB ? BA. A matrix multiply- ing or multiplied by the identity I
MATRICES AND MATRIX OPERATIONS
26 fév. 2018 MULTIPLYING MATRICES – We can multiply only matrices where the first matrix has the number of columns same as the number of rows of the second ...
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[PDF] Lecture 2 Matrix Operations - EE263
Lecture 2 Matrix Operations • transpose sum difference scalar multiplication • matrix multiplication matrix-vector product • matrix inverse 2–1
[PDF] Basic Matrix Operations - George Brown College
A matrix is a rectangular or square grid of numbers arranged into rows and columns Each number in the matrix is called an element and they are arranged in
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26 fév 2018 · MULTIPLYING MATRICES – We can multiply only matrices where the first matrix has the number of columns same as the number of rows of the second
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In this appendix some basic concepts of matrix algebra necessary for formu- To illustrate the procedure of matrix multiplication we compute the prod-
[PDF] Matrix Operations and Their Applications
Matrix Operations and Their Applications The dimension of a matrix is defined as a pair of numbers representing the number of rows and columns that a
[PDF] 75 Operations with Matrices
Add and subtract matrices and multiply matrices by scalars • Multiply two matrices • Use matrix operations to model and solve real-life problems
[PDF] Matrices and Matrix Operations
where aij is the number corresponding to the ith row and jth column i is the row subscript and j is the column subscript – The size of the matrix is mxn
[PDF] Matrices
For example the matrices above have dimensions 2 × 3 3 × 3 and 1 × 4 Basic Matrix Operations Addition (or subtraction) of matrices is performed by
[PDF] Matrices
Basic Matrix Operations Addition (or subtraction) of matrices is performed by adding (or subtracting) elements in corresponding positions
[PDF] Chapter 2 Matrices
We write down some of the properties of matrix addition and multiplication 2 We define transpose of a matrix 3 We start writing some proofs Page 12
Matrices and Matrix Operations
Linear Algebra
MATH 2010
Basic Denition and Notation for Matrices
{Ifmandnare positive integers, then anmxnmatrixis a rectangular array of numbers (entries) mrows8 >>:2 6 664a11a12a13::: a1n
a21a22a23::: a2n............
a m1am2am3::: amn3 7 775|{z} ncolumns whereaijis the number corresponding to theithrow andjthcolumn.iis the row subscript and jis the column subscript. {Thesizeof the matrix ismxn. {Matrices are denoted by capital letters:A,B,C, etc. {Ifm=n, then the matrix is said to besquare. {For a square matrix,a11,a22,a33, ...,annis called themain diagonal. {tr(A) denotes thetraceofAwhich is the sum of the diagonal elements. For example, if A=2
43 5 2
01 43 1 23
5 then the diagonal elements are 3, -1, and 2, so tr(A) = 3 + (1) + 2 = 4 {Acolumn vectoris a matrix with only 1 column, i.e., it has sizemx1. Example: 2 413 23
5 {Arow vectoris a matrix with only 1 row, i.e., it has size 1xn. Example:
01 2 8
{Two matrices areequalif they are the same size and all the entries are the exact same. For example, if A=2 4 1 3 andB=a4 1b what doaandbhave to equal forA=B? Adding and Subtracting Matrices: IMPORTANT!!! In order to add/subtract matrices, matrices must be theSAMEsize. If two matrices are the same size, then to add (subtract) them, we simply add (subtract) corresponding elements. LetA=2 1 1
11 4 andB=23 4 3 12 ThenA+B=2 1 1
11 4 +23 43 12
2 + 2 1 + (3) 1 + 4
1 + (3)1 + 1 4 + (2)
42 54 0 2 and
AB=2 1 1
11 4 23 43 12
22 1(3) 14
1(3)11 4(2)
0 43 22 6Scalar Multiplication:Example, 2A. In order to do scalar multiplication, multiply all entries by the scalar. For example, using the matrixAfrom above, i.e.,
A=2 1 1
11 4 we can calculate 2Aas2A=2(2) 2(1) 2(1)
2(1) 2(1) 2(4)
=4 2 2 22 8Linear Combination:IfA1,A2, ...,Anare matrices of the same size andc1,c2, ...,cnare scalars, then c
1A1+c2A2+:::+cnAn
is called alinear combinationofA1,A2, ...,Anwith coecientsc1,c2, ...,cn. For example, ifA=2 1 1
11 4 ; B=23 4 3 12 ; C=0 51 1 04 then2A3B+C= 22 1 1
11 4 323 43 12 +0 51 1 04
2(2) 2(1) 2(1)
2(1) 2(1) 2(4)
3(2) 3(3) 3(4)
3(3) 3(1) 3(2)
+0 51 1 04 4 2 2 22 869 12
9 36 +0 51 1 04
46 + 0 2(9) + 5 212 + (1)
2(9) + 123 + 0 8(6) + (4)
2 1611
85 10Properties of Matrix Addition and Substraction:LetA,BandCbemxnmatrices andcand dbe scalars, then
1.A+B=B+ACommutative Property of Addition
2.A+ (B+C) = (A+B) +CAssociative Property of Addition
3. (cd)A=c(dA) Associative Property of Scalar Multiplication
4. 1A=AMultiplicative Identity
5.c(A+B) =cA+cBDistributive Property
6. (c+d)A=cA+dADistributive Property
Transposes:
{The transpose of a matrix is denotedAT. To nd the transpose of a matrix, you interchange the rows and columns. In other words, you can think about it as write all the rows as columns or all the columns as rows. For example, if A=1 2 31then A T=1 3 21
Find the transpose of
A=2 41 20 5 323
5 {Notice that ifAismxn, thenATisnxm. {Some Properties of Transposes
1. (AT)T=A
2. (A+B)T=AT+BT
3. (cA)T=cAT
{A matrix is said to besymmetricifA=AT.Sample Problems:
1. Let
A=21 5
0 3 5FindAT.
2. Let
A=241 2 3
0 5 4 32 135 andB=2 431
0 1 5 43 5
Find (A+ (2B)T)T.
3. Findcanddso that
A=241 2 5
2 3c d4 03 5 is symmetric.Special Matrices:There are two special matrices,
{ Identity Matrixis denoted byIorInwherendenotes a square matrix of sizenxn. The identitity matrix is a square matrix with 1's on the diagonal and 0's as all other elements: I 2=1 0 0 1 ; I 4=2 6641 0 0 0
0 1 0 0
0 0 1 0
0 0 0 13
7 75{ Zero Matrix Thezero matrixis denoted byOorOmxnwhereOis a matrix of sizemxn. This is simply a matrix with all zeros. Example: O=0 0 0 0 ;orO=2 40 0
0 0 0 03 5
Properties of the Zero Matrix
1.A+O=Awhere it is understood thatOhas the same size asA.
2.A+ (A) =O
3. IfcA=O, thenc= 0 orA=O.
Matrix Multiplication:Matrix multiplication is more involved. You canNOTmultiply correspond- ing entries!! {To help understand the process of matrix multiplication, we will rst examine an applied problem which uses the same strategy as is used in matrix multiplication. Assume you are at a football stadium where there are three dierent refreshment centers, the south stand, north stand and west stand. At each stand, they are selling peanuts, hot dogs and soda. See the gure below. Assume you want to know how much total the south stand made. You need to multiply the number of each of the items sold by the south stand (in the rst row of the matrix) by the selling price of each item (given in the column vector containing selling price). In other words, you need to isolate the rst row and multiply by the corresponding items in the column and then add:120 250 3052
42:003:00 2:753 5 = 120(2:00) + 250(3:00) + 305(2:75) = 1828:75:
So, the south stand sold a total of $ 1828.75.
Similarly, the north stand sold
207 140 4192
42:003:00 2:753 5 = 207(2:00) + 140(3:00) + 419(2:75) = 1986:25:
And the west stand sold
29 120 1902
42:003:00 2:753 5 = 29(2:00) + 120(3:00) + 190(2:75) = 940:50: What we just did was matrix multiplication. We multiplied a 3x3 matrix by a 3x1 matrix to get a 3x1 matrix: 2
4120 250 305
207 140 419
29 120 1903
5242:00
3:00 2:753 5 =2
41828:75
1986:25
940:53
5 { Size of matrices is important!Notice above, that we multiplied two matrices together, one was size 3x3 and the other was size 3x1. They are NOT the same size. LetAbe amxnmatrix andBbe apxqmatrix. In order to multiplyAB,the number of columns ofAmust equal thenumber of rows ofB. The schematic below will help.So, ifAbe amxnmatrix andBbe apxq, then in order to multiplyAB,nmust equalpand the
resulting size ofABismxq. { Examples:First, determine if it is possible to ndABandBAby looking at the sizes of the matrix. If so, what is the size of the resulting matrix? Find the resulting matrix.1.A=1 2
2 1 ,B=32 4 22.A=1 2
3 4 ,B=01 2 33 4 0 1
3.A=2 42 13 4 1 63 5 ,B=2 401 0
4 0 2 81 73
5 { Properties of Matrix Multiplication:LetA,B, andCbe matrices of appropriate size for matrix multiplication andcbe a scalar, then the following properties hold.
1.A(BC) = (AB)CAssociative property of multiplication
2.A(B+C) =AB+ACDistributive property
3. (A+B)C=AC+BCDistributive property
4.c(AB) = (cA)B=A(cB)
{ Commutativity:In general,AB6=BA!. Note that ifABis dened,BAmay not be dened. { Cancelation:IfAC=BC, you can NOT sayA=B. You can not simply cancel like in scalar multiplication. There are conditions onCwhich must be met in order to apply the cancelation principle. We will discuss these conditions in a later section. As an example, given A=241 2 3
0 5 4 32 135 ;B=2 446 3
5 4 4
1 0 13
5 ;C=240 0 0
0 0 0 42 335quotesdbs_dbs20.pdfusesText_26
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