18.06.04: ‘Matrices’
Lecturer: Barwick
Wednesday 10 February 2016
18.06.04: ‘Matrices’
Figure 1: What are the angles between my bonds?
You were saying?
18.06.04: ‘Matrices’
Exam 1 is a week from today
▶ It should be pretty simple.
▶ I will cover the course material through Friday’s lecture.
▶ You may not use any aids.
18.06.04: ‘Matrices’
The projection of a vector 𝑏⃗ onto a vector 𝑎⃗ is the vector
⃗
⃗ 𝑎 = 𝑎⃗ ⋅ 𝑏 𝑎.⃗
𝜋𝑎⃗ (𝑏)⃗ ≔ (̂
𝑎 ⋅ 𝑏)̂
𝑎⃗ ⋅ 𝑎⃗
𝑎⃗
.
𝑏⃗
𝜋𝑎⃗ (𝑏)⃗
18.06.04: ‘Matrices’
Matrices
Matrices are rectangular arrays of real numbers:
𝑎11 𝑎12
𝑎
𝑎22
𝐴 = ( 21
⋮ ⋮
𝑎𝑚1 𝑎𝑚2
⋯ 𝑎1𝑛
⋯ 𝑎2𝑛
)
⋱ ⋮
⋯ 𝑎𝑚𝑛
18.06.04: ‘Matrices’
Here, 𝐴 has 𝑚 rows and 𝑛 columns. We say that 𝐴 is an 𝑚 × 𝑛 matrix. We can
think of 𝐴 as a sequence of 𝑛 vectors in R𝑚 :
𝐴 = ( 𝐴1⃗ 𝐴2⃗ ⋯ 𝐴𝑛⃗ )
or as a sequence of 𝑚 row vectors in R𝑛 :
𝐴⃗ 1
𝐴⃗
𝐴 = ( 2 ).
⋮
𝐴⃗ 𝑚
18.06.04: ‘Matrices’
The first thing that makes the notion of a matrix interesting is that you can
multiply matrices by vectors. If
𝑏1
𝑏
𝑏⃗ = ( 2 ) ∈ R𝑛 ,
⋮
𝑏𝑛
then we can write
𝑎11 𝑏1 + 𝑎12 𝑏2 + ⋯ + 𝑎1𝑛 𝑏𝑛
𝑎 𝑏 + 𝑎22 𝑏2 + ⋯ + 𝑎2𝑛 𝑏𝑛
𝐴𝑏⃗ = 𝑏1 𝐴1⃗ + 𝑏2 𝐴2⃗ + ⋯ + 𝑏𝑛 𝐴𝑛⃗ = ( 21 1
)
⋮
𝑎𝑚1 𝑏1 + 𝑎𝑚2 𝑏2 + ⋯ + 𝑎𝑚𝑛 𝑏𝑛
18.06.04: ‘Matrices’
Or, equivalently,
𝐴⃗ 1 ⋅ 𝑏⃗
𝑎11 𝑏1 + 𝑎12 𝑏2 + ⋯ + 𝑎1𝑛 𝑏𝑛
⃗
⃗
𝐴 ⋅𝑏
𝑎 𝑏 + 𝑎22 𝑏2 + ⋯ + 𝑎2𝑛 𝑏𝑛
𝐴𝑏⃗ ≔ ( 2
) = ( 21 1
)
⋮
⋮
𝐴⃗ 𝑚 ⋅ 𝑏⃗
𝑎𝑚1 𝑏1 + 𝑎𝑚2 𝑏2 + ⋯ + 𝑎𝑚𝑛 𝑏𝑛
18.06.04: ‘Matrices’
One way to say what’s going on here is to say that multiplication by an 𝑚 × 𝑛
matrix 𝐴 is a function
𝑇𝐴 ∶ R𝑛
R𝑚
that is defined so that
𝑇𝐴 (𝑥)⃗ = 𝐴𝑥.⃗
This is an important way to think about matrices, and so we should pay close
attention to this picture.
18.06.04: ‘Matrices’
Let’s think about this function applied to the unit vectors 𝑒𝑖̂ :
𝑇𝐴 (𝑒𝑖̂ ) = 𝐴𝑒𝑖̂ = ?
18.06.04: ‘Matrices’
𝑎11 𝑎12
𝑎
𝑎22
𝑇𝐴 (𝑒𝑖̂ ) = 𝐴𝑒𝑖̂ = ( 21
⋮ ⋮
𝑎𝑚1 𝑎𝑚2
(
⋯ 𝑎1𝑛
(
(
⋯ 𝑎2𝑛
)(
(
(
⋱ ⋮
(
(
⋯ 𝑎𝑚𝑛
(
0
0
⋮
0
1
0
⋮
0
𝐴 = ( 𝐴𝑒1̂ 𝐴𝑒1̂ ⋯ 𝐴𝑒𝑛̂ )
)
)
)
)=(
)
)
)
)
)
𝑎1𝑖
𝑎2𝑖
) = 𝐴𝑖⃗ .
⋮
𝑎𝑚𝑖
18.06.04: ‘Matrices’
Question. If 𝜃 ∈ [0, 2𝜋), then we can form this 2 × 2 matrix:
𝑅𝜃 ≔ (
cos 𝜃 − sin 𝜃
).
sin 𝜃 cos 𝜃
Describe the corresponding function 𝑇𝑅𝜃 ∶ R2
R2 geometrically.
18.06.04: ‘Matrices’
Armed with this, we can compile systems of 𝑚 linear equations
𝑣1
=
𝑎11 𝑥1 + 𝑎12 𝑥2 + ⋯ + 𝑎1𝑛 𝑥𝑛 ;
𝑣2
=
𝑎21 𝑥1 + 𝑎22 𝑥2 + ⋯ + 𝑎2𝑛 𝑥𝑛 ;
⋮
𝑣𝑚
=
𝑎𝑚1 𝑥1 + 𝑎𝑚2 𝑥2 + ⋯ + 𝑎𝑚𝑛 𝑥𝑛 ;
into the single equation 𝑣 ⃗ = 𝐴𝑥,⃗ where 𝑣 ⃗ ∈ R𝑛 is a fixed vector, and 𝑥⃗ ∈ R𝑚 is
a variable vector. Solving the system is solving the compiled equation.
18.06.04: ‘Matrices’
But so what? It’s the same data, but a different format.
The point is that qualitative things about the system of linear equations can
be extracted from qualitative things about the matrix.
18.06.04: ‘Matrices’
Here’s a diagonal 𝑛 × 𝑛 matrix:
𝜆1 0
0 𝜆2
𝐴 = diag (𝜆1 , 𝜆2 , … , 𝜆𝑛 ) ≔ (
⋮ ⋮
0 0
⋯ 0
⋯ 0
);
⋱ ⋮
⋯ 𝜆𝑛
𝜆1 𝑏1
𝜆 𝑏
𝐴𝑏⃗ = ( 2 2 ) ,
⋮
𝜆𝑛 𝑏𝑛
so equations 𝐴𝑥⃗ = 𝑣 ⃗ will be wicked easy to solve (uniquely??).
18.06.04: ‘Matrices’
Only slightly less easy to solve will be what we get out of an 𝑛 × 𝑛 upper
triangular matrix:
𝑎11 𝑎12
0 𝑎22
𝐴=(
⋮ ⋮
0
0
⋯ 𝑎1𝑛
⋯ 𝑎2𝑛
)
⋱ ⋮
⋯ 𝑎𝑛𝑛
𝑎11 𝑏1 + 𝑎12 𝑏2 + ⋯ + 𝑎1𝑛 𝑏𝑛
𝑎 𝑏 + 𝑎23 𝑏3 + ⋯ + 𝑎2𝑛 𝑏𝑛
𝐴𝑏⃗ = ( 22 2
).
⋮
𝑎𝑛𝑛 𝑏𝑛
18.06.04: ‘Matrices’
Question. Suppose 𝐴 an 𝑚 × 𝑛 matrix. When is a vector in R𝑚 of the form
𝐴𝑏⃗ for some vector 𝑏⃗ ∈ R𝑛 ?
Hint: We gave this formula
𝐴𝑏⃗ = 𝑏1 𝐴1⃗ + 𝑏2 𝐴2⃗ + ⋯ + 𝑏𝑛 𝐴𝑛⃗ ,
which exhibits 𝐴𝑏⃗ as a linear combination of the column vectors of 𝐴 …
18.06.04: ‘Matrices’
Question. Suppose 𝐴 an 𝑚 × 𝑛 matrix. When is the system of linear equations
𝐴𝑥⃗ = 𝑣 ⃗ redundant? That is, when is it the case that the equations provide the
same constraints on the variable 𝑥⃗ that could be provided with fewer equations?
Hint: we have the dual formula
𝐴⃗ 1 ⋅ 𝑏⃗
𝐴⃗ ⋅ 𝑏⃗
𝐴𝑏⃗ ≔ ( 2
)
⋮
𝐴⃗ 𝑚 ⋅ 𝑏⃗