FP3 Chapter 6 �Further Matrix Algebra
www.dilanmaths.com
What are Matrices again?
You probably remember a variety of operations to do with matrices: multiplying, inversing, etc.
But what is the point of a matrix?
The formal definition of a matrix is that it’s just ‘a grid of numbers’.
But its main purpose is describe how we get from one coordinate space to another, e.g. rotating the coordinate axis, enlarging, shearing or even creating/discarding dimensions.
Examples:
?
?
?
?
?
Rows and Columns of Matrices
Each row of a matrix is an instruction of how to generate each dimension of the new coordinate space…
It’s therefore best perhaps to visualise the effect of a matrix by where the new coordinate axes end up, as this determines the effect on all transformed points.
Further Example
Click for Willmanimation
?
Matrix Transpose
(Beyond understanding required for exam)
Symmetric and Identity Matrices
?
Example
?
?
?
Further Properties of Transposes
?
?
(We’ll skip Exercise 6A, but you’re welcome to practise this in your own time)
(note the minus for the middle one)
?
Test Your Understanding
?
Exercise 6B
Recall that we swap the number in this diagonal.
…and negate the remaining numbers.
We divide by the determinant.
?
?
?
The method we used was a specific case of a more general method which can be used for matrices of any size:
A cofactor by definition is ‘a signed minor’. We simply apply signs to each minor using the following alternating pattern: (+ top left)
?
?
Bro Tip: Note we’ve already found the top row from above.
?
?
?
?
Doing with your silver calculator
Test Your Understanding
Bro Exam Tip: I couldn’t find a single exam question where you were asked to invert a matrix without there being a variable involved.
?
?
We can get rid of each matrix on the front by multiplying both sides by the inverse.
?
?
?
Exercise 6C
Matrices for Transformations
Working out the Matrix for a Transformation
The matrices go right-to-left, i.e. if we multiply by a matrix, it goes on the front.
?
?
?
Another Example
?
?
?
Transforming a Line/Plane
?
?
Test Your Understanding
?
Reminder of forms of plane equation
?
?
(Part of June 2014 question)
Last Example
?
Exercise 6D
Questions in textbook.
Bonus Exam Questions:
?
?
Using inverse matrix to reverse transformation
Bro Exam Note: So far, no questions have come up on this skill.
?
Exercise 6E
Eigenvectors and Eigenvalues
?
?
?
?
?
Why do we care about eigenvectors?
?
Identifying these ‘principal axes’ is crucial in physics, e.g. analysing inertia of a rigid body.
Example: Dimensionality Reduction
Suppose that we wanted to represent this 2D data in just 1D.
This might be because there’s a ‘redundant’ dimension that doesn’t yield interesting info about the data.
We might for example by interested how far along the blue line each data point is, because in the direction perpendicular to this, the data doesn’t vary very much, and could be down to ‘noise’/errors in the data.
This is particularly important when we have a large number of dimensions (e.g. various measurements from a person’s face: nose length, width between eyes, etc.), but want to analyse the data in a smaller dimensional space, for example to more easily determine someone’s gender based on these measurements.
What we want…
(Just for fun)
Example: Dimensionality Reduction
As per previous discussion, the eigenvectors (of the covariance matrix) give us the ‘axes’.
What matrix could we therefore use to rotate the data in such a way that the eigenvectors end up at the coordinate axes?
(Just for fun)
Now back to A Level…
🖉 A normalised eigenvector is one what is written as a unit vector.
Allowed because matrix multiplication is distributive.
Example
Using definition of eigenvector.
Use characteristic equation.
?
In 3D…
Broculator Tip: You may want to use the simultaneous equation solver on your silver calculator.
?
Test Your Understanding
?
Invariant Line
This is known as an invariant line, because any point on the line will be mapped to another point on the line.
a
b
?
One final quick example
a
b
?
Exercise 6F
Decomposing a Matrix
LU (‘Lower Upper’) Decomposition
Matrix decomposed into a ‘lower triangular’ and ‘upper triangular’ matrix.
(The names should be obvious from the image)
Eigendecomposition
This is the decomposition method we’ll be covering…
But why?
🖉
Eigendecomposition
Transformation 1
‘Orthogonal’ just means perpendicular.
What transformation does this represent?
A rotation such that the eigenvectors are moved to the coordinate axis (recall that we do the inverse because we want the eigenvectors to move to the coordinate axis, not vice versa).
?
Transformation 2
?
?
🖉
Eigendecomposition
Transformation 3
?
🖉
Eigendecomposition
Enlargement
Enlargement
?
?
🖉
Let’s take some notes…
An exam question once had a 2 marker getting you to prove this rearrangement.
Example
?
?
?
?
?
?
?
?
?
?
Another Example
?
Test Your Understanding
? a
Test Your Understanding
? b
Test Your Understanding
? c
Further Test Your Understanding (the last ever)
(c) on next slide.
Bro Note: Notice they’ve given us the eigenvectors, but not the original matrix. This is why there’s part (c), where we’re forced to do some actual matrix multiplication!
? a
? b
Further Test Your Understanding (the last ever)
Bro Note: Notice they’ve given us the eigenvectors, but not the original matrix. Hence part (c), where we’re forced to do some actual matrix multiplication!
? c
Exercise 6G
You have reached the end of maths.*
* At A Level.