Singular matrices... does a matrix always have an inverse?
• No! Matrices need not have an inverse
• If a matrix does not have an inverse, it is not invertible, or it is
called singular
Consider our expression for M −1 ,
M −1 =
1
CT
det M
• M is singular if det M = 0
Geometric interpretation:
If we consider the rows (or columns) of M to be vectors (consider
a 3 × 3 matrix, the determinant is the volume Ω of a polyhedron
described by the vectors. Then Ω = det M = 0 corresponds to the
case where the three vectors lie all in a plane. This is related to
the question of linear dependence/independence that we will
address later.
Patrick K. Schelling
Introduction to Theoretical Methods
Rotation matrix in two dimensions
• Linear transformations are important applications of linear
algebra
• Coordinate transformations are important, including rotations
• Start with a vector ~r = x î + y ĵ and rotate by angle φ
~
• After rotation, x → x 0 and y → y 0 , or ~r → R
• We can find the rotation matrix R(φ) and the transformation
0 x
cos φ − sin φ
x
=
0
y
sin φ cos φ
y
• What is the inverse of this matrix? Think of the simplest idea
Patrick K. Schelling
Introduction to Theoretical Methods
Inverse of rotation matrix... rotate back by angle −φ!
• We could rotate back with R −1 (φ) = R(−φ)
cos φ − sin φ
R(φ) =
sin φ cos φ
cos φ sin φ
−1
R (φ) = R(−φ) =
− sin φ cos φ
• Try it!
R
−1
(φ)R(φ) =
cos φ sin φ
− sin φ cos φ
Patrick K. Schelling
cos φ − sin φ
sin φ cos φ
=
Introduction to Theoretical Methods
1 0
0 1
Multiple rotations, and another (equivalent) picture
• It is easy to show that R(θ)R(φ) = R(θ + φ) (see Problem 25,
section 6)
• Moreover, R(γ)R(θ)R(φ) = R(γ + θ + φ), etc.
• In another picture, we can imagine rotating the axes by an angle
φ, and leaving vector ~r = x î + y ĵ fixed
• In this picture, x 0 and y 0 define point in new coordinate system,
and are given by
0 x
cos φ sin φ
x
=
y0
− sin φ cos φ
y
• Both of these pictures are often used... In one rotating the
vector, and the other rotating the coordinate system
• Rotation is example of a linear operator describing a linear
transformation
• It is also an example of an orthogonal transformation
Patrick K. Schelling
Introduction to Theoretical Methods
Linear combinations, functions, and operators
I
~ +B
~
A linear combination is simply addition of two vectors, A
Patrick K. Schelling
Introduction to Theoretical Methods
Linear combinations, functions, and operators
I
~ +B
~
A linear combination is simply addition of two vectors, A
I
A scalar function of a vector is f (~r )
Patrick K. Schelling
Introduction to Theoretical Methods
Linear combinations, functions, and operators
I
~ +B
~
A linear combination is simply addition of two vectors, A
I
A scalar function of a vector is f (~r )
~ (~r )
A vector function of a vector is F
I
Patrick K. Schelling
Introduction to Theoretical Methods
Linear combinations, functions, and operators
I
~ +B
~
A linear combination is simply addition of two vectors, A
I
A scalar function of a vector is f (~r )
~ (~r )
A vector function of a vector is F
I
I
An operator acts on a scalar, vector, etc. and results in a new
scalar, vector, etc.
Patrick K. Schelling
Introduction to Theoretical Methods
Examples: Linear functions and operators
• To be a linear function, we require that f (~r1 +~r2 ) = f (~r1 ) + f (~r2 )
and that f (a~r ) = af (~r ) where a is a scalar.
• Example: A scalar function of a vector, problem 3, section 7
Is the following scalar function of the vector ~r linear?
f (~r ) = ~r · ~r
We see that for ~r = ~r1 + ~r2 ,
f (~r ) = f (~r1 + ~r2 ) = (~r1 + ~r2 ) · (~r1 + ~r2 ) = ~r1 · ~r1 + ~r2 · ~r2 + 2~r1 · ~r2
But, for f (~r1 ) + f (~r2 )
f (~r1 ) + f (~r2 ) = ~r1 · ~r1 + ~r2 · ~r2
Thus it is not linear. The other requirement also is not met.
Patrick K. Schelling
Introduction to Theoretical Methods
Examples: Linear functions and operators
• To be a linear function, we require that
~ (~r1 + ~r2 ) = F
~ (~r1 ) + F
~ (~r2 ) and that F
~ (a~r ) = aF
~ (~r ) where a is a
F
scalar.
• Example: A vector function of a vector, problem 5, section 7
~ is a given
Is the following function of the vector ~r linear? The A
vector.
~ (~r ) = A
~ × ~r
F
~ × (~r1 + ~r2 ) = A
~ × ~r1 + A
~ × ~r2 , and also
It is obvious that A
~ × (a~r ) = aA
~ × ~r
A
~
~ × ~r is a linear vector function
Therefore, F (~r ) = A
Patrick K. Schelling
Introduction to Theoretical Methods
Linear operators
• A linear operator and the properties O(A + B) = O(A) + O(B)
and O(kA) = kO(A), where k is just a number
d
• Example: Is dx
a linear operator?
df
dg
d
[f (x) + g (x)] =
+
dx
dx
dx
d
df
kf (x) = k
dx
dx
d
So yes, dx
is a linear operator.
• The elements A and B above can be numbers, functions,
vectors, etc.
• In quantum mechanics we consider operators in some vector
space, continuous or discrete
Patrick K. Schelling
Introduction to Theoretical Methods
Orthogonal transformation
• Consider a transformation that does not change the length of the
vector. This is an orthogonal transformation
(x 0 )2 + (y 0 )2 = x 2 + y 2
• Rotation good example of orthogonal transformation
x0
0 2
0 2
0
0
(x ) + (y ) = x y
y0
We just use,
x0
y0
x0
y0
x
=
=
y
cos φ sin φ
− sin φ cos φ
cos φ − sin φ
sin φ cos φ
x
y
• Note that R(φ) is an example of a linear operator
Patrick K. Schelling
Introduction to Theoretical Methods
Continued...orthogonal transformation
Then we get for
x
y
(x 0 )2
+
(y 0 )2
cos φ sin φ
− sin φ cos φ
=
x0
y0
x0
y0
cos φ − sin φ
sin φ cos φ
,
x
y
= x 2 +y 2
This is obvious because we just have R −1 (φ)R(φ),
cos φ sin φ
cos φ − sin φ
1 0
=
− sin φ cos φ
sin φ cos φ
0 1
Patrick K. Schelling
Introduction to Theoretical Methods
Continued...orthogonal transformation
• We also observe that R −1 (φ) = R T (φ).
R
−1
(φ) =
cos φ sin φ
− sin φ cos φ
=
cos φ − sin φ
sin φ cos φ
T
= R T (φ)
• For any orthogonal transformation M, we find M T = M −1
• We also find for orthogonal transformations det M = ±1
We can show this since M T M = I , and then
det (M T M) = (det M T )(det M) = (det M)2 = 1
• If det M = 1, purely a rotation
• If det M = −1, either a reflection/inversion
Patrick K. Schelling
Introduction to Theoretical Methods
Inversions/reflections
• Another example of orthogonal transformation
• Simplest example, reflection across x = 0 (x → −x and y → y )
−1 0
M=
0 1
• Here we see detM = −1 as expected for pure reflection
• M −1 = M T = M (Try it!)
−1 0
−1 0
1 0
MT M =
=
=I
0 1
0 1
0 1
Patrick K. Schelling
Introduction to Theoretical Methods
More general inversions
• We might want to invert across some line other than x = 0 or
y = 0.
• There is a straightforward way to do this combining rotation of
coordinate axes, inversion, and then another rotation back to
original system
• Determine the transformation matrix M for inversion across the
line y = −x
First rotate axes by an angle φ = − π4 . Next inversion across new
axis y = 0. Finally rotate coordinate axes back by φ = π4 .
!
! −1
√1
√1
√1
√
1
0
0 −1
2
2
2
2
=
M=
√1
√1
0 −1
−1 0
− √12 √12
2
2
This matrix has M T = M −1 = M and det M = −1 for reflections.
Further check: M should leave a vector −î + ĵ unchanged
−1
−1
0 −1
=
−1 0
1
1
Patrick K. Schelling
Introduction to Theoretical Methods
Determining what transformation a matrix represents
• We should be able to figure out what kind of transformation a
matrix M represents
• If det M = 1, M is a rotation, but if det M = −1, M is a
reflection (If det M is not ±1, it is not an orthogonal
transformation)
• Example: Problem 23, section 7:
Determine if the matrix M is orthogonal, and whether it is a
rotation or reflection, and lastly find the rotation angle or line of
reflection
!
√
1
− 23
2√
M=
− 21 − 23
√
1
− 23
2√ = 1. This corresponds
First, we see that det M = −1 − 3 2 √
to a rotation. We have cos φ = −
φ = 7π
6 .
Patrick K. Schelling
3
2
2
and sin φ = − 12 . Hence,
Introduction to Theoretical Methods
Another example...
• Example: Problem 26, section 7:
Determine if the matrix M is orthogonal, and whether it is a
rotation or reflection, and lastly find the rotation angle or line of
reflection
3
4
5
5
M=
4
3
5 −5
3
4 5 = −1, so this is a reflection.
First we see that det M = 54
3 −
5
5
To find the line of the reflection, find a vector that is unchanged
by the transformation.
3
4
x
x
5
5
=
4
3
y
y
−
5
5
We can also write this as,
2
−5
4
5
4
5
− 85
Patrick K. Schelling
x
y
=0
Introduction to Theoretical Methods
Problem 26 continued...
− 52
4
5
4
5
− 85
x
y
=0
Both of the equations this represents are identical, so we find easily
that y = 12 x, so a line 2î + ĵ describes the line of reflection.
Patrick K. Schelling
Introduction to Theoretical Methods