Matrices, part 2: Matrix Multiplication

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Motivation for Definition of Matrix Multiplication

Supposev1, . . . ,vnis a basis ofV

andw1, . . . ,wmis a basis ofW.

Supposeu1, . . . ,upis a basis ofU.

Consider linear mapsT : U → V andS: V → W. Does M(ST) equal M(S)M(T)? Suppose M(S) = A and M(T) = C. For1 ≤ k ≤ p, we have (ST)uk=S n X r=1 Cr,kvr = n X r=1 Cr,kSvr = n X r=1 Cr,k m X j=1 Aj,rwj = m X j=1 n X r=1 Aj,rCr,kwj.

Thus M(ST) is the m-by-p matrix whose entry in row j, column k, is

n

X

r=1

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Motivation for Definition of Matrix Multiplication

n

X

r=1

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Motivation for Definition of Matrix Multiplication

n

X

r=1

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Motivation for Definition of Matrix Multiplication

n

X

r=1

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Motivation for Definition of Matrix Multiplication

n

X

r=1

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Definition of Matrix Multiplication

Definition:matrix multiplication

SupposeA is an m-by-n matrix and C is an n-by-p matrix. Then AC is defined to be them-by-p matrix whose entry in row j, column k, is given by the following equation:

(AC)j,k = n

X

r=1

Aj,rCr,k.

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Definition of Matrix Multiplication

(AC)j,k = n

X

r=1

Aj,rCr,k.

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Definition of Matrix Multiplication

(AC)j,k = n

X

r=1

Aj,rCr,k.

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Matrices and Products of Linear Maps

In the following result, the same basis ofV is used in considering T ∈ L(U, V) and S ∈ L(V, W), the same basis of W is used in

consideringS ∈ L(V, W) and ST ∈ L(U, W), and the same basis of U is used in consideringT ∈ L(U, V) and ST ∈ L(U, W).

The matrix of the product of linear maps

IfT ∈ L(U, V) and S ∈ L(V, W), then M(ST) = M(S)M(T).

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Notation for Row and Column of Matrix

Notation: A_j,· ,A·,k

SupposeA is an m-by-n matrix.

If1 ≤ j ≤ m, then A_j,·denotes the1-by-n matrix consisting of rowj of A.

If1 ≤ k ≤ n, then A·,k denotes them-by-1 matrix consisting of

columnk of A. Example: IfA = 8 4 5 1 9 7

, thenA2,·is row2 of A and A·,2is

column2 of A. In other words,

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Notation for Row and Column of Matrix

Notation: A_j,· ,A·,k

SupposeA is an m-by-n matrix.

If1 ≤ j ≤ m, then A_j,·denotes the1-by-n matrix consisting of rowj of A.

If1 ≤ k ≤ n, then A·,k denotes them-by-1 matrix consisting of

columnk of A. Example: IfA = 8 4 5 1 9 7

, thenA2,·is row2 of A and A·,2is

column2 of A. In other words,

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Alternative Ways to Think about Matrix Multiplication

Example: 3 4 6 2 = 26 = 26.

Entry of matrix product equals row times column

SupposeA is an m-by-n matrix and C is an n-by-p matrix. Then (AC)j,k =Aj,·C·,k

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Alternative Ways to Think about Matrix Multiplication

Example: 3 4 6 2 = 26 = 26.

Entry of matrix product equals row times column

SupposeA is an m-by-n matrix and C is an n-by-p matrix. Then (AC)j,k =Aj,·C·,k

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Alternative Ways to Think about Matrix Multiplication

Column of matrix product equals matrix times column

SupposeA is an m-by-n matrix and C is an n-by-p matrix. Then (AC)·,k=AC·,k

for1 ≤ k ≤ p.

Row of matrix product equals row times matrix

SupposeA is an m-by-n matrix and C is an n-by-p matrix. Then (AC)j,· =Aj,·C

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Alternative Ways to Think about Matrix Multiplication

Column of matrix product equals matrix times column

SupposeA is an m-by-n matrix and C is an n-by-p matrix. Then (AC)·,k=AC·,k

for1 ≤ k ≤ p.

Row of matrix product equals row times matrix

SupposeA is an m-by-n matrix and C is an n-by-p matrix. Then (AC)j,· =Aj,·C

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Matrix Times a Column Vector

Linear combination of columns

SupposeA is an m-by-n matrix and c =    c1 .. . cn   is ann-by-1 matrix. ThenAc = c1A·,1+ · · · +cnA·,n.

In other words,Ac is a linear combination of the columns of A, with the scalars that multiply the columns coming fromc.

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Matrix Times a Column Vector

Linear combination of columns

SupposeA is an m-by-n matrix and c =    c1 .. . cn   is ann-by-1 matrix. ThenAc = c1A·,1+ · · · +cnA·,n.

In other words,Ac is a linear combination of the columns of A, with the scalars that multiply the columns coming fromc.

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Row Vector Times a Matrix

Linear combination of rows

Supposea = a1 · · · an is a1-by-n matrix and C is an n-by-p

matrix. Then

aC = a1C1,·+ · · · +anCn,·.

In other words,aC is a linear combination of the rows of C, with the scalars that multiply the rows coming froma.

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Row Vector Times a Matrix

Linear combination of rows

Supposea = a1 · · · an is a1-by-n matrix and C is an n-by-p

matrix. Then

aC = a1C1,·+ · · · +anCn,·.

In other words,aC is a linear combination of the rows of C, with the scalars that multiply the rows coming froma.

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