dm qm background mathematics lecture 10

8/10/2019 Dm Qm Background Mathematics Lecture 10

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Quantum Mechanics for

Scientists and Engineers

David Miller


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Background mathematics 10

Linear equations and matrices


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Linear equations and matrices

Suppose we have equations for two

straight linesNote, if you are used to the form

we can rewrite this as

so these are equivalentWe can rewrite these equations as

the one matrix equation

or, with b1 instead of x and b2instead of y

in summation form

11 12 1

12 22 2

A x A y c

A x A y c

y mx c

11 12 1 12( / ) ( / )y A A x c A

11 12 1

21 22 2

A cx xAA A cy y

2

1

mn n m

n

A b c


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Linear equation solutions

With the linear equations in matrix

formwe can formally solve them if weknow the inverse

multiplying by

Sincewe have the solution

the intersection point of

the linesSolving linear equations and invertinga matrix are the same operation

11 12 1

21 22 2

A A cx x

A A A cy y

1A

1A 11 1

2

cxA A Acy

1 A I

11

2

cx Acy

( , )x y


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Determinant

If the determinant of a matrix is

not zerothen the matrix has an inverse

and if a matrix has an inverse,

the determinant of the matrixis not zero

A nonzero determinant is a

necessary and sufficient

condition for a matrix to beinvertible


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Determinant of a matrix

The determinant of a matrix is

written in one of two notations

There are two complete formulasfor calculating it

Leibnizs formulaLaplaces formula

and many numerical techniquesto calculate it

we will not give these generalformulas or methods here

11 12 1

21 22 2

1 2

det

N

N

N N NN

A A

A A AA

A A A

A


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Determinant of a 2x2 matrix

For a 2x2 matrix

we add the product on the leadingdiagonal

and subtract the product on the otherdiagonal

11 12 11 22 12 2121 22

detA A

A A A A AA A


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Determinant of a 3x3 matrix

For a 3x3 matrix, we have

11 22 33 23 3211 12 13

21 22 23 12 21 33 23 31

31 32 3313 21 32 22 31

det

A A A A AA A A

A A A A A A A A A

A A A A A A A A

+ +-11

2

12 13

21 2 23

32 3331

A

A

A

A

A A

A

A

11 13

2

12

21 22

331 332

3

A

A A

A

A

A A

A

A

13

21 2

11 12

22

31 32

3

33

A

A A

A A

A A

A

A


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General form of determinant

If we multiply out the 3x3 determinant expression

we see that each term, e.g., contains adifferent element from each row

and the elements in each term are never fromthe same column

but we always have one element from eachrow and each column in each term

11 22 33 23 32 12 21 33 23 31 13 21 32 22 3111 22 33 11 23 32 12 21 33 12 23 31 13 21 32 13 22 31

det A A A A A A A A A A A A A A A A

A A A A A A A A A A A A A A A A A A

12 23 31A A A


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General form of the determinant

We see that this form

contains every possible term with one

element from each rowall from different columns

this is a general property ofdeterminants

11 22 33 11 23 32 12 21 33 12 23 31 13 21 32 13 22 31det A A A A A A A A A A A A A A A A A A A


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General form of the determinant

To understand how to construct a

determinantit only remains to find the sign of the terms

To do so

count the number of adjacent row (orcolumn) swaps required

to get all the elements in the term ontothe leading diagonal

if that number is even, the sign is +if that number is odd, the sign is -


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Sign of determinant terms

For the term

we have to perform 1 row swap

1 is an odd number

so the sign of this term in the

determinant is negative

11 23 32A A A

12 13

21 2

11

232

31 3 332

A

A A

A

A A

A

A

11

32

23

12 13

31 33

21 22

A A

A

A A

A

A

A


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Sign of determinant terms

For the term

we have to perform 2 row swaps

2 is an even number

so the sign of this term in the

determinant is positive

This is actually Leibnizs determinant formula

12 23 31A A A

11 13

21 2

12

23

31

2

32 33

A

A

A

A

A

A

A 31

12

23

32 33

11 13

21 22

A

A

A A

A

A

A

11 13

32 33

1

21 2

2

1

22

3

3

A A

A A

A A

A

A


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Background mathematics 10

Eigenvalues and eigenvectors


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Matrix eigenequation

An equation of the form

where d is a vector, is a number, and isa square matrix

is called aneigenequation

with eigenvalue

and eigenvector d

If there are solutions

they may only exist for specific values of

Ad dA


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Solving an eigenequation

We can rewrite as

where we have introduced the identitymatrix

which we can always do becauseSo

(strictly, the 0 here is a vector with elements 0)

so, writing we have

d d

Ad Id

I

Id d

0A I d

B A I 0Bd


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Now, for to have any solutions for any

non-zero vector dthe matrix cannot have an inverse

if it did have an inverse

but any (finite) matrix multiplying azero vector must give a zero vector

so there is no non-zero solution d

Hence, by reductio ad absurdum,has no inverse

0Bd

B1B

1 1 0B Bd Id d B

B


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The fact that has no inverse means

from the properties of the determinant

This equation will allow us to constructa secular equation

whose solutions will give theeigenvalues

From those we will deduce thecorresponding eigenvectors d

B A I

det 0A I


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Suppose we want to find the eigenvalues and

eigenvectorsif they exist

of the matrix

So we write the determinant condition forfinding eigenvalues

1.5 0.5 1 0

det det 00.5 1.5 0 1

i

A I i

1.5 0.50.5 1.5

iA

i


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Now

So our secular equation becomes, from

i.e.,

1.5 0.5 1 0 1.5 0.5 0det det

0.5 1.5 0 1 0.5 1.5 0

1.5 0.5

det 0.5 1.5

i i

i i

i

i

2 2

1.5 (0.5 ) 0.5 1.5 0.25 0i i

2 3 2 0

det 0A I


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Solving this quadratic equation

gives rootsand

Now that we know the eigenvalues

we substitute them back into theeigenequation

and deduce the correspondingeigenvectors

2 3 2 0

1 1 2 2


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Our eigenequation is, explicitly

where now, for a given eigenvalue

we are trying to find d1 and d2so we know the corresponding eigenvector

Rewriting gives

Ad d

1 1

2 2

1.5 0.5

0.5 1.5

d di

d di

1

2

1.5 0.5 00.5 1.5 0

didi


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Now evaluating

for a specific eigenvalue

say, the first one,

gives

or, as linear equations

1

2

1.5 0.5 0

0.5 1.5 0

di

di

1 1

1

2

0.5 0.5 0

0.5 0.5 0

di

di

1 2

1 2

0.5 0.5 00.5 0.5 0

d id

id d


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Solving eigenequations

For larger matrices with eigensolutions

e.g.,we have correspondingly higher order

polynomial secular equations

which can have Neigenvalues andeigenvectors

e.g., a matrix can have 3 eigenvalues andeigenvectors

Note eigenvectors can be multiplied by anyconstant and still be eigenvectors

N N

3 3


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dm qm background mathematics lecture 10

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