Dm Qm Background Mathematics Lecture 9

8/10/2019 Dm Qm Background Mathematics Lecture 9

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Quantum Mechanics for

Scientists and Engineers

David Miller


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Background mathematics 9

Matrix notation


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Matrix notation

A matrix is, first of all, a rectangulararray of numbers

AnMNmatrix has

Mrows (here 2)

Rows are horizontalNcolumns (here 3)

Columns are vertical

The array is enclosed in square brackets

2 1 36 5 4

A

This is arectangular

matrix

2 3


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Symbol for a matrix

As a symbol for a matrix

we could just use a capital letter, likeA

Here, we need to distinguish matrices

and other linear operators

from numbers and simple variablesso we put a hat over a symbol

representing a matrix

which distinguishes a matrixsymbol when we write it byhand

A

2 1 36 5 4

A


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Rectangular and square matrices

Because all matrices are, bydefinition, rectangular

when we say a matrix isrectangular

we almost always mean it is nota square matrix

one with equal numbers ofrows and columns

This is asquarematrix

2 2

1.5 0.5

0.5 1.5

iB

i


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Rectangular and square matrices

The numbers or elements in a matrixcan be

real, imaginary, or complex

The elements are indexed in row-

column orderB12 is the element (value -0.5i) in the

first row and second column

We often use the same letter, hereB, forthe matrix and for its elements

or the lower case version, e.g., b12


2 2

1.5 0.5

0.5 1.5

iB

i


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Diagonal elements

The leading diagonal of a matrix

or just the diagonal

is the diagonal from top left tobottom right

Elements on the diagonalhere those with value 1.5

are called diagonal elements

Elements not on the diagonalhere those with value 0.5i and -0.5i

are called off-diagonal elements


2 2

1.5 0.5

0.5 1.5

iB

i


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Vectors

In the matrix algebra version of vectors

which are matrices of size 1 in one oftheir directions

we must specify whether a vector is a

row vectora matrix with one row

or a column vector

a matrix with one column

2 3

5 2

4

7 6

i

i

d i

i

4, 2,5, 7c


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Transpose

An important manipulation for matricesand vectors is

the transpose

denoted by a superscript T

a reflection about a diagonal linefrom top left to bottom right fora matrix

Algebraically

2 1 36 5 4

A

2 6 1 5

3 4

TA

T nmmn

A A


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1.5 0.50.5 1.5

iBi

Transpose


the transpose



Algebraically

1.5 0.50.5 1.5

T i

Bi

T nmmn

B B


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Transpose


the transpose



or at 45 for a vector

4, 2,5,7c

42

5

7

Tc


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Transpose


the transpose



or at 45 for a vector

2 3

5 2

4

7 6

i

i

d i

i

[2 3 5 2 4 7 6 ]Td i i i i


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Hermitian transpose or adjoint

Another common manipulation is the

Hermitian adjoint, Hermitiantranspose, or conjugate transpose

denoted by a superscript

pronounced daggera reflection about a diagonal line

from top left to bottom right for amatrix or at 45 for a vector

and taking the complexconjugate of all the elements

1.5 0.50.3 1.5

iDi

1.5 0.3

0.5 1.5

iD

i

nmmnD D


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Hermitian transpose or adjoint

Another common manipulation is the

Hermitian adjoint, Hermitiantranspose, or conjugate transpose

denoted by a superscript

pronounced daggera reflection about a diagonal line

from top left to bottom right for amatrix or at 45 for a vector

and taking the complexconjugate of all the elements

2 3

5 2

4

7 6

i

i

d i

i

2 3 5 2 4 7 6d i i i i


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Hermitian matrix

A matrix is said to be

Hermitian

if it is equal to its own Hermitianadjoint

i.e.,

or, element by element

B B

nm nmB B

1.5 0.50.5 1.5

iBi

1.5 0.5 0.5 1.5

iB B

i


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Background mathematics 9

Matrix algebra


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Adding and subtracting matrices

If two matrices are the same

sizei.e., the same numbers ofrows and columns

we can add or subtractthem by

adding or subtractingthe individual matrix

elementsone by one

1

2 1 3

iF

i

5 4

6 7 8

iG

i

K F G


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elementsone by one

1

2 1 3

iF

i

5 4

6 7 8

iG

i

1 5 42 6 1 7 3 8

K F G

i ii i


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elementsone by one

1

2 1 3

iF

i

5 4

6 7 8

iG

i

1 5 42 6 1 7 3 8

6 5

4 8 5

K F G

i i

i i

i

i


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Multiplying a vector by a matrix

Suppose we want to multiply a column

vector by a matrixThe number of rows in the vector

must match the number ofcolumns in the matrix

This is generally true for matrix-matrixmultiplication

The number of rows in the matrix onthe right

must match the number ofcolumns in the matrix on the left

1 2 3

4 5 6

matrix

7

8

9

vector


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First we put the vector sideways on

top of the matrixthen multiply element by element

and add to get the first element

of the resulting vectorMove down

and repeat for the next row

1 2 3

4 5 6

7

8

9

matrix vector

1 2 3

4 5 6

7 8 91 7

2 8

3 950

50

1 2 3

4 5 67 8 9

4 7

5 86 9

122

50122


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1 2 3

4 5 6

7

8

9

matrix vector

7

8

9

1 2 3

4 5 6

50

122

=

m mn n

n

d A c

d A c

First we put the vector sideways on

top of the matrixthen multiply element by element

and add to get the first elementof the resulting vector

Move down

and repeat for the next row

We can also write this multiplication

with a sum over the repeatedindex


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Multiplying a matrix by a matrix

To multiply a matrix by a matrix

repeat this operation for eachcolumn of the matrix on theright

working from left to right

Write down the resultingcolumns in the resulting matrix

also working from left to right

Summation notationsums over the repeated index

7 150 14 1 2 3

8 2122 32 4 5 6

9 3

R B A

mp mn np

nR B A


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Vector vector products

An inner product

of a row and a column vectorcollapses two vectors to a

number

analogous to geometricalvector dot product

An outer product

of a column and a row vector

generates a square matrix

4

32 1 2 3 5

6

4 8 12 4

5 10 15 5 1 2 3

6 12 18 6

n n

n

f c d c df

mp m pF d c

F d c


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Matrix algebra properties

Matrix algebra, like normal algebra

is associative

and has distributive properties

but matrix multiplication is

not in general commutativeas is easily proved byexample

CB A C BA A B C AB AC

1 2 5 6 19 22

3 4 7 8 43 50

5 6 1 2 23 347 8 3 4 31 46

BA AB in general


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Multiplying a matrix by a number


means we multiply every element ofthe matrix by that number

Also, we can take out a common factor

from every elementmultiplying the matrix by that factor

Such results are easily proved insummation notation

e.g., for matrix vector multiplicationwhereBmn= Amn

1 2 2 42

3 4 6 8

2 4 1 2

26 8 3 4

m mn n mn n

n n

mn n mn n

n n

d A c A c

A c B c


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Since number multiplication is commutative

we can move simple factors aroundarbitrarily in matrix products

e.g., for a number and a vector c

This result is also easily proved usingsummation notation

ABc A Bc AB c


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Hermitian adjoint of a product

The Hermitian adjoint of a product

is the reversed product of the Hermitian adjoints

We can prove this using summation notation

Suppose so that and so

AB B A

R AB mp mn npn

R A B

*

( ) ( )

( ) ( ) ( ) ( )

pm mp mn np mn npn npm

nm pn pn nmn n pm

AB R R A B A B

A B B A B A


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Inverse of a matrix

For ordinary algebra, the reciprocal or

inverse of a number or variablex is

which has the obvious property

For a matrix, if it has an inverseit has the property

where is called the identity matrix

which is the diagonal matrix with1 for all diagonal elements

and zeros for all other elements

1 1x

x xx

1/x 1xor

1

1 A A I I


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Identity matrix

For example, the 3x3 identity matrix is

The identity matrix in a givenmultiplication has to be the right size

so we do not typically bother to state

the size of the identity matrixFor any matrix

we can write

Like the number 1 in ordinary algebra

the identity matrix is almost trivial

but is very important

1 0 0

0 1 0

0 0 1

I

AI IA A

A


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Dm Qm Background Mathematics Lecture 9