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Multiplication by scalars: if A is a matrix of size m × n and c is a scalar, then cA is a matrix of size m × n Matrix multiplication: if A is a matrix of size m × n and B is a matrix of size n × p, then the product AB is a matrix of size m × p

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Inverting Matrices P Danziger Matrix Algebra Theorem 3 (Algebraic Properties of Matrix Multiplication) 1 (k + l)A = kA + lA (Distributivity of scalar multiplication 

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from the fact that the sum of two matrices is only defined when they have the same size Properties of matrix multiplication (MM1): The product (AB)C is defined 

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18 fév 2002 · matrix addition and scalar multiplication, is a vector space 2 Properties of Matrix Multiplication and In- verse Matrices Theorem 5 Let A, B and 

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2 nov 2005 · The number of columns of A must be equal to the number of rows of B If A is 2 × n and B is n × 1 then AB is 2 × 1 This rule for multiplication may be extended to matrices, A, which have more than two rows

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B Properties of Matrix Multiplication: Theorem 1 2 Let A, B, and C be matrices of appropriate sizes Then the following properties hold: a) A(BC)=(AB)C 

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Scalar multiplication: to multiply a matrix A by a scalar r, one That is, matrices are multiplied row by column: ( ∗ ∗ ∗ Properties of matrix multiplication:

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(A + B)C = AC + BC and A(B + C) = AB + AC Note: Matrix-matrix multiplication does not commute Only in very rare cases does AB equal BA Indeed, the 

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Section 2.4 - Properties of Matrix-Matrix

Multiplication

Maggie Myers

Robert A. van de Geijn

The University of Texas at Austin

Practical Linear Algebra { Fall 2009

http://z.cs.utexas.edu/wiki/pla.wiki/1

Matrix-Matrix Multiplication is Associative

LetA,B, andCbe matrices of conforming dimensions. Then (AB)C=A(BC):Proof

Letejequal thejth unit basis vector. Then

(AB)Cej= (AB)cj=A(Bcj) =A(BCej) =A(BC)ej: Thus, the columns of(AB)Cequal the columns ofA(BC), making the two matrices equal. http://z.cs.utexas.edu/wiki/pla.wiki/2

Matrix-Matrix Multiplication is Distributive

LetA,B, andCbe matrices of conforming dimensions. Then (A+B)C=AC+BCandA(B+C) =AB+AC:Note: Matrix-matrix multiplication does not commute Only in very rare cases doesABequalBA. Indeed, the matrix dimensions may not even be conformal. http://z.cs.utexas.edu/wiki/pla.wiki/3

Theorem

LetA2RmkandB2Rkn. Then

(AB)T=BTAT:Lemma LetA2Rmn. TheneTiAT= (Aei)TandATej= (eTjA)T.Proof of Lemma The proof of this lemma is pretty obvious: Theith row ofATis clearly theith column ofA, but viewed as a row, etc.http://z.cs.utexas.edu/wiki/pla.wiki/4 Proof of Theorem that(AB)T=BTATWe prove that the(i;j)element of(AB)Tequals the(i;j) element of(BTAT): e

Ti(AB)Tej

=<(i;j)element ofCequals(j;i)element ofCT> e

Tj(AB)ei

= (eTjA)(Bei) =< xTy=yTx > (Bei)T(eTjA)T = equotesdbs_dbs20.pdfusesText_26