CMSE/MTH 314 Matrix Algebra with Computational Applications

• exam code

• Course website - D2L
• Lecture:
  • Instructor: Son Tu – tuson@msu.edu
  • Wells Hall A332 (TR 9:10 am - 12:00 pm).
• Syllabus PDF
• Schedule PDF

Office hours:

• Wednesday 9-11 am on Zoom or in person C330 Wells Hall
• By appointment (via email)
• https://msu.zoom.us/j/96240880040
• Meeting ID: 962 4088 0040
• Passcode: 632835

Getting help:

• MLC - Math Learning Center
• Piazza - MTH314
• MSU Counseling Center
• Student Resources
• Resource Center for Persons with Disabilities

Materials

• All materials are hosted on D2L.
• Some lecture notes are posted here for convenience.

• CMSE/MTH 314 Matrix Algebra with Computational Applications
• Materials
  • Lecture 11   ·   2024-06-18   ·   Diagonalization & Singular value decomposition
  • Lecture 10   ·   2024-06-13   ·   Eigenvalue & Markov's problem
  • Lecture 09   ·   2024-06-11   ·   Gram-Schmidt process
  • Lecture 08   ·   2024-06-06   ·   Change of basis
  • Lecture 07   ·   2024-06-04   ·   Vector space 2
  • Lecture 06   ·   2024-05-30   ·   Vector space I
  • Lecture 05   ·   2024-05-28   ·   Linear transformation
  • Lecture 04   ·   2024-05-23   ·   Determinant
  • Lecture 03   ·   2024-05-21   ·   Inverse of Matrix

Lecture 11   ·   2024-06-18   ·   Diagonalization & Singular value decomposition

• Diagonalization
• Singular value decompostion
  • Theorem: If $A$ is of size $n\times n$ is symmetric and orthogonal, then $A$ can be written as $A = PDP^T$, where $P$ is orthogonal and $D$ is diagonal.
  If $A$ is of size $m\times n$, then * the matrix $A^TA$ is $n\times n$ symetric * the matrix $AA^T$ is $m\times m$ symetric We therefore can apply the theorem to $A^TA$ and $AA^T$ to get $$ \begin{cases} A^TA &= VD_1V^T \\ AA^T &= UD_2U^T . \end{cases} $$ Let $\lambda_1, \ldots, \lambda_n$ be the eigenvalues of $A^TA$, then they are non-negative. The singular values are $\sigma_i=\sqrt{\lambda_i}$ for $i=1,2,\ldots,n$. Let $D = \mathrm{diag}(\sigma_1,\ldots,\sigma_n)$ then $$ A = U\cdot \Sigma \cdot V^T $$ where $\Sigma$ is the $m\times n$ matrix fromed from $D$ with padded zeros rows and columns.
  • Notes: when computing $U,V$, we need to normalized every eigenvector (column) to norm $1$.
  • Facts for orthogonal matrix $A$
    • Column vectors are an orthogonal set
    • Row vectors are an orthogonal set
    • $A^{-1}$ is orthorgonal
    • $A^T$ is orthorgonal (since $A^{-1} = A^T$)

Lecture 10   ·   2024-06-13   ·   Eigenvalue & Markov's problem

Lecture 09   ·   2024-06-11   ·   Gram-Schmidt process

Lecture 08   ·   2024-06-06   ·   Change of basis

Key takeaways:

• Let $\mathcal{E} = (e_1,\ldots, e_n)$ be a (standard) basis for $V$ (think of $\R^n$)
• Let $\mathcal{B} = (b_1,\ldots, b_n)$ be a basis for $V$.
• Let $\mathcal{C} = (c_1,\ldots, c_n)$ be a basis for $V$.

For $x\in V$ then

$$ [x]_\mathcal{B} = \begin{bmatrix} x_1\\ \vdots\\ x_n \end{bmatrix} \qquad\text{means} \qquad x =[x]_\mathcal{E} = x_1b_1 + \ldots + x_nb_n. $$

In other words, this is the key equation:

$$ [ x ] _ { \mathcal{E} } = [ b_1, \ldots, b_n ] \cdot [ x ] _ {\mathcal{B}} = [c_1, \ldots, c_n] \cdot [ x ] _ {\mathcal{C}}. $$

Then we can convert between bases as

$$ \fbox{$ \displaystyle [ x ] _ { \mathcal{C} } = [ c_1,\ldots, c_n ] ^ {-1} \cdot [b_1,\ldots, b_n] \cdot [ x ] _ { \mathcal{B} } . $} $$

Lecture 07   ·   2024-06-04   ·   Vector space 2

Key takeaways: Given a matrix $A$ of size $m\times n$

• Nullspace: set of all solutions of the equation $Ax=0$.
  Other notation: $\mathrm{null}(A)$ or $\mathrm{ker}(T)$ where $T(x)=Ax$.
• Column space: set of linear combinations of the columns of $A$.
  
  • $\mathrm{col}(A)$ is a subset of $\R^m$
  • $\mathrm{col}(A)$ is also range of $T$ where $Tx=Ax$
• Row space: set of linear combinations of all rows of $A$ (each row is a vector in $\R^n$)
  • $\mathrm{row}(A) = \mathrm{col}(A^T)$
• Finding bases for row space, column space, null space and dimension of a space
  • Row space: rwo equivalent transformation
    Start from $A$, do RREF, whatever we get at the end, their rows form a basis for $\mathrm{row}(A)$
  • Note: row operation $A$ to $B$ makes $\mathrm{col}(A)=\mathrm{col}(B)$ (as vector spaces)
  • Column space:
    • Start from $A$, do RREF to get $B$
    • See which columns of $B$ are dependent on the others (non-pivot columns), cross them out
    • The rest of the columns form a basis for $\mathrm{col}(A)$
• $\mathrm{rank}(A) = \mathrm{dim}(\mathrm{col}(A)) = \mathrm{dim}(\mathrm{row}(A))$ : number of leading (pivots) in RREF
• Rank-Nullity theorem:
  • $\mathrm{rank}(A) + \mathrm{dim}(\mathrm{null}(A)) = n$
  • $\mathrm{rank}(A) + \mathrm{dim}(\mathrm{null}(A^T)) = m$

Lecture 06   ·   2024-05-30   ·   Vector space I

Lecture 05   ·   2024-05-28   ·   Linear transformation

Key takeaways:

• Linear transformation (they are matrix tranformation if $T:\R^n\to\R^m$)
• Given a linear transformation $T:\R^n\to\R^m$, find the matrix representation $Tx=Ax$.
• Elementary transformations in 2D:
  • Shears (vertically, horizontally)
  • Projecttions (to $x_1$ axis, to $x_2$ axis)
  • Contractions and Expansions (vertically, horizontally)
  • Reflections (5 types)
  • Rotations
• A special nonlinear transformation: translation.
• How to combine transformation (the order matters).

Lecture 04   ·   2024-05-23   ·   Determinant

Key takeaways:

• Algorithm to compute determinant: cofactor expansion
• Relation between determinants of matrices after row operations.

import numpy as np
A = np.array([[1,-4,2], [-2,8, -4], [-1, 7, 0]])
det = np.linalg.det(A)

Lecture 03   ·   2024-05-21   ·   Inverse of Matrix

Key takeaways:

• Finding the inserve of a matrix.
• Finding a series of elementary matrices in the row operations.

import numpy as np
import sympy as sym

A = sym.Matrix([
    [0, 1, 2],
    [1, 0, 3],
    [4,-3, 8]
])

# stack A|B to form the augmented matrix
AB = np.column_stack((A,np.eye(3)))

# convert AB to sympy
AB = sym.Matrix(AB)

# perform the row operations to find the reduced echelon form of A
AB_rref = AB.rref()[0]

# select the matrix on the right hand side
Ainv = AB_rref[:, 3:6]

Ainv

© 2016-2024 Son N. T. Tu. Powered by Ark. Last updated: October 02, 2024.