Ch 3.3. Square matrices, determinants, and trace

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1. Square matrices

• A square matrix $\text{M}$ has dimensions $n\times n$. (Thus, $\mathbf{M}\in {\mathbb{R}}^{n\times n}$)
• Square matrices linearly transform all vectors in an ${\mathbb{R}}^n$ vector space to ${\mathbb{R}}^n$ vector space (i.e. no dimensionality change).
  • This means the vector space is transformed in a way that no dimensions are added or lost.
  • Also, the origin remains fixed, and all grid lines remain parallel and evenly spaced.

2. Determinant, $\text{det}(\text{M})$ or $|\text{M}|$

• The Determinant, $\text{det}(\text{M})$ or $|\text{M}|$, is an important property of square matrices.

2.1. Geometric intuition (From 3Blue1Brown):

• Its absolute value, $|\text{det}(\text{M})|$, tells you the factor by which any and all regions in the vector space increases/decreases after the transformation by matrix $\text{M}$

• A “region” can be thought of as depending on the dimensionality of the vector space:

  • For a 2D vector space, a “region” is an area (e.g. of the unit square)
  • For a 3D vector space, a “region” is a volume (e.g. of the unit cube)
  • And so on…

• How to interpret the determinant in terms of initial region (area/volume/etc) in the vector space (before transformation):

  • Absolute values:
    • If $|\text{det}(\text{M})|>1$: Any/all initial regions in the vector space are scaled up by that factor
    • If $|\text{det}(\text{M})|=1$: Any/all initial regions in the vector space do not change in size
    • If $0<|\text{det}(\text{M})|<1$: Any/all initial regions in the vector space shrink by that factor
  • If $\text{det}(\text{M})=0$: it means space has dropped into fewer dimensions (i.e. information lost)
    • i.e. at least 1 dimension is LOST, so all initial regions (area/volume/etc) of the vector space is now 0
    • e.g. if a 2D matrix $\text{M}$ transforms a 2D vector space into a 1D line (or a 0D point), all “areas” in the original 2D vector space no longer exist. They are squished to 0; hence $\text{det}(\text{M})=0$
    • This also means that the columns of $\text{M}$ are linearly dependent (i.e. they lie on the same line/plane/etc)
  • If $\text{det}(\text{M})<0$
    • A negative sign tells you whether the vector space was “flipped” (i.e. if the basis vectors swapped sides).
    • Here, $\text{M}$ is said to “invert the orientation” of the vector space.

• To prove understanding, explain: $\text{det}({\text{M}}_1{\text{M}}_2)=\text{det}({\text{M}}_1)\text{det}({\text{M}}_2)$

2.2. Formulae for calculating the determinant of a matrix $\text{M}$:

• For a $2\times 2$ matrix; $\text{det}(\text{M})$ is:

$$\begin{array}{r}|\text{M}|=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]=ad-bc\end{array}$$

• For a $3\times 3$ matrix; $\text{det}(\text{M})$ is:

$$\begin{array}{r}\begin{array}{rlr}|\text{M}|=\left[\begin{array}{ccc}a& b& c\\ d& e& f\\ g& h& i\end{array}\right]& =a\left[\begin{array}{ccc}\square & \square & \square \\ \square & e& f\\ \square & h& i\end{array}\right]-b\left[\begin{array}{ccc}\square & \square & \square \\ d& \square & f\\ g& \square & i\end{array}\right]+c\left[\begin{array}{ccc}\square & \square & \square \\ d& e& \square \\ g& h& \square \end{array}\right]& \\ & & \\ & =a\left[\begin{array}{cc}e& f\\ h& i\end{array}\right]-b\left[\begin{array}{cc}d& f\\ g& i\end{array}\right]+c\left[\begin{array}{cc}d& e\\ g& h\end{array}\right]& \\ & & \\ & =aei+bfg+cdh-ceg-bdi-afh& \end{array}\end{array}$$

• For higher dimension matrices, a similar approach can be used.

2.2.1 Geometric derivation of the $2\times 2$ determinant formula:

• This relies on how the square matrix $\text{M}$ transforms the unit square in ${\mathbb{R}}^2$ into a parallelogram in ${\mathbb{R}}^2$.
• See A Few Useful Transformations section for a recap of how each element of the matrix $\text{M}$ affects the transformation of the unit square.

2.3. Identity matrix

• A square matrix, $\text{I}$, with ones on the diagonal and zeros everywhere else
• Multiplying a matrix with $\text{I}$ (of compatible dimensionality) will produce the same matrix (like how $n\times 1=n$)

# Find the determinant of M, and multiply M by np.eye(4) to demonstrate M x I = M
 
import numpy as np
from numpy.linalg import det
 
M = np.array([[0, 2, 1, 3], [3, 2, 8, 1], [1, 0, 0, 3], [0, 3, 2, 1]])
print("M:\n", M)
 
print(f"Determinant: {det(M):.1f}")
I = np.eye(4)
print("\nI:\n", I)
print("\nM @ I:\n", M @ I)
 

M:
 [[0 2 1 3]
 [3 2 8 1]
 [1 0 0 3]
 [0 3 2 1]]
Determinant: -38.0
 
I:
 [[1. 0. 0. 0.]
 [0. 1. 0. 0.]
 [0. 0. 1. 0.]
 [0. 0. 0. 1.]]
 
M @ I:
 [[0. 2. 1. 3.]
 [3. 2. 8. 1.]
 [1. 0. 0. 3.]
 [0. 3. 2. 1.]]

3. Trace

• The trace of $\text{A}:\text{A}\in {\mathbb{R}}^{n\times n}$ is the sum of elements on the main diagonal (from left to right):

$\text{tr}(\text{A})={\sum }_{i=1}^n{a}_{ii}$

# Compute the trace of the following matrix A
 
from numpy import trace
 
A = np.array([[4.1, 2.8], [9.7, 6.6]])
print("Trace(A)", trace(A))
 

Trace(A) 10.7

3. 3Blue1Brown’s aside

A few useful transformations

Vector Transformations

Matrix Transformations (TBC double check)

$$\begin{array}{rlr}\text{No change (Identity matrix)}& :\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Translation}& :\text{NA}& \text{in 3D:}\left[\begin{array}{ccc}1& 0& X\\ 0& 1& Y\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Scaling (about origin)}& :\left[\begin{array}{cc}W& 0\\ 0& H\end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}W& 0& 0\\ 0& H& 0\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Rotation (about origin)}& :\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}\cos \theta & -\sin \theta & 0\\ \sin \theta & \cos \theta & 0\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Shear (in x-direction)}& :\left[\begin{array}{cc}1& \tan \varphi \\ 0& 1\end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}1& \tan \varphi & 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Shear (in y-direction)}& :\left[\begin{array}{cc}1& 0\\ \tan \psi & 1\end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}1& 0& 0\\ \tan \psi & 1& 0\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Reflect (about origin)}& :\left[\begin{array}{cc}-1& 0\\ 0& -1\end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}-1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Reflect (about x-axis)}& :\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}1& 0& 0\\ 0& -1& 0\\ 0& 0& 1\end{array}\right]\\ & & \\ \text{Reflect (about y-axis)}& :\left[\begin{array}{cc}-1& 0\\ 0& 1\end{array}\right]& \text{in 3D:}\left[\begin{array}{ccc}-1& 0& 0\\ 0& 1& 0\\ 0& 0& 1\end{array}\right]\\ & & \end{array}$$