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Lecture 2

Matrix Operations

transpose, sum & difference, scalar multiplication matrix multiplication, matrix-vector product matrix inverse 2-1

Matrix transpose

transposeofmnmatrixA, denotedATorA, isnmmatrix with AT ij=Aji rows and columns ofAare transposed inAT example: 0 4 7 0 3 1 T =0 7 34 0 1 transpose converts row vectors to column vectors, vice versa ATT=A

Matrix Operations2-2

Matrix addition & subtraction

ifAandBare bothmn, we formA+Bby adding corresponding entries example: 0 4 7 0 3 1 1 2 2 3 0 4 1 6 9 3 3 5 can add row or column vectors same way (but never to each other!) matrix subtraction is similar:

1 69 3

I=0 69 2

(here we had to figure out thatImust be22)

Matrix Operations2-3

Properties of matrix addition

commutative:A+B=B+A associative:(A+B)+C=A+(B+C), so we can write asA+B+C

A+ 0 = 0 +A=A;AA= 0

(A+B)T=AT+BT

Matrix Operations2-4

Scalar multiplication

we can multiply a number (a.k.a.scalar) by a matrix by multiplying every entry of the matrix by the scalar this is denoted by juxtaposition or, with the scalar on the left: (2) 1 6 9 3 6 0 212
186
12 0 (sometimes you see scalar multiplication with the scalar onthe right) (α+β)A=αA+βA;(αβ)A= (α)(βA)

α(A+B) =αA+αB

0A= 0;1A=A

Matrix Operations2-5

Matrix multiplication

ifAismpandBispnwe can formC=AB, which ismn C ij=p k=1a ikbkj=ai1b1j++aipbpj, i= 1,...,m, j= 1,...,n to formAB, #cols ofAmust equal #rows ofB; calledcompatible to findi,jentry of the productC=AB, you need theith row ofA and thejth column ofB form product of corresponding entries,e.g., third component ofith row ofAand third component ofjth column ofB add up all the products

Matrix Operations2-6

Examples

example 1:

1 69 3

01 1 2 =6 11 33
for example, to get1,1entry of product: C

11=A11B11+A12B21= (1)(0) + (6)(1) =6

example 2: 01 1 2 1 6 9 3 =93 17 0 these examples illustrate that matrix multiplication is not (in general) commutative: we don"t (always) haveAB=BA

Matrix Operations2-7

Properties of matrix multiplication

0A= 0,A0 = 0(here0can be scalar, or a compatible matrix)

IA=A,AI=A

(AB)C=A(BC), so we can write asABC

α(AB) = (αA)B, whereαis a scalar

A(B+C) =AB+AC,(A+B)C=AC+BC

(AB)T=BTAT

Matrix Operations2-8

Matrix-vector product

very important special case of matrix multiplication:y=Ax

Ais anmnmatrix

xis ann-vector yis anm-vector y i=Ai1x1++Ainxn, i= 1,...,m can think ofy=Axas a function that transformsn-vectors intom-vectors a set ofmlinear equations relatingxtoy

Matrix Operations2-9

Inner product

ifvis a rown-vector andwis a columnn-vector, thenvwmakes sense, and has size11,i.e., is a scalar: vw=v1w1++vnwn ifxandyaren-vectors,xTyis a scalar calledinner productordot productofx,y, and denotedx,yorxy: x,y=xTy=x1y1++xnyn (the symbolcan be ambiguous - it can mean dot product, or ordinary matrix product)

Matrix Operations2-10

Matrix powers

if matrixAis square, then productAAmakes sense, and is denotedA2 more generally,kcopies ofAmultiplied together givesAk: A k=A AA k by convention we setA0=I (non-integer powers likeA1/2are tricky - that"s an advanced topic) we haveAkAl=Ak+l

Matrix Operations2-11

Matrix inverse

ifAis square, and (square) matrixFsatisfiesFA=I, then

Fis called theinverseofA, and is denotedA1

the matrixAis calledinvertibleornonsingular ifAdoesn"t have an inverse, it"s calledsingularornoninvertible by definition,A1A=I; a basic result of linear algebra is thatAA1=I we define negative powers ofAviaAk=A1k

Matrix Operations2-12

Examples

example 1: 11 1 2 1 =13 2 1 1 1 (you should check this!) example 2: 11 2 2 does not have an inverse; let"s see why: a b c d 11 2 2 =a2ba+ 2b c2dc+ 2d =1 00 1 . . . but you can"t havea2b= 1anda+ 2b= 0

Matrix Operations2-13

Properties of inverse

A11=A,i.e., inverse of inverse is original matrix

(assumingAis invertible) (AB)1=B1A1(assumingA,Bare invertible)

AT1=A1T(assumingAis invertible)

I1=I (αA)1= (1/α)A1(assumingAinvertible,α= 0) ify=Ax, wherexRnandAis invertible, thenx=A1y: A

1y=A1Ax=Ix=x

Matrix Operations2-14

Inverse of22matrix

it"s useful to know the general formula for the inverse of a22matrix: a b c d 1 =1 adbc db c a providedadbc= 0(ifadbc= 0, the matrix is singular) there are similar, but much more complicated, formulas for the inverse of larger square matrices, but the formulas are rarely used

Matrix Operations2-15

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