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MAT223 Lecture 06

MAT223 Lecture 06 Raw
MAT223 Lecture 06 Flashcards

• Completed Notes Status

  • Completed insertions: 3
  • Ambiguities left unresolved: 0

• Lecture Summary

  • Central objective: Explore the properties of commutativity in matrix multiplication and the characteristics of symmetric and skew-symmetric matrices.
  • Key concepts:
    • Commutativity: Matrix multiplication is not commutative in general ($AB\ne BA$). However, specific matrices can commute, such as the identity matrix or powers of the same matrix. We can solve for a matrix $B$ that commutes with a given $A$ by setting up a System of Linear Equations.
    • Symmetry Properties:
      • A matrix is Symmetric if $A^T=A$.
      • A matrix is Skew-Symmetric if $A^T=-A$.
      • For any square matrix $A$, $A+A^T$ is symmetric and $A-A^T$ is skew-symmetric.
    • Matrix-Vector Products: We can represent a linear combination of the columns of a matrix $A$ as the product $A\overrightarrow{x}$.
  • Connections:
    • Solving for commuting matrices ($AB=BA$) reduces to solving a homogeneous System of Linear Equations derived from the entry-wise equations.
    • Commutability requires compatible dimensions: if $A$ is $m\times n$ and $B$ is $k\times l$, both products exist only if $n=k$ and $l=m$.

• TK Resolutions

  • #tk: "Looks like it is true" (referring to the skew-symmetry of $A-A^T$).
    • Answer: The raw note immediately provided a "General Argument" verifying that $(A-A^T{)}^T=A^T-A=-(A-A^T)$, proving the skew-symmetry formally. No further action is needed.

• Practice Questions

  • Remember/Understand:
    • If $A$ is $3\times 5$, what must the dimensions of $B$ be for both $AB$ and $BA$ to be defined?
    • What is the defining property of a skew-symmetric matrix involving its transpose?
  • Apply/Analyze:
    • Given $A=\left[\begin{array}{cc}0& 1\\ 0& 0\end{array}\right]$, find all matrices $B$ such that $AB=BA$.
    • Prove that for any square matrix $A$, the matrix $C=A+A^T$ is symmetric.
  • Evaluate/Create:
    • Construct a $2\times 2$ matrix $B$ such that $AB\ne BA$ for $A=\left[\begin{array}{cc}1& 1\\ 1& 1\end{array}\right]$.

• Challenging Concepts

  • Commuting Matrices:
    • Why it's challenging: Students often assume commutativity holds because it does for real numbers, or they struggle to set up the system of equations to find commuting matrices.
    • Study strategy: Explicitly write out the products $AB$ and $BA$ with variables for the unknown matrix, then equate corresponding entries to form a system.
  • Skew-Symmetric Matrices:
    • Why it's challenging: The diagonal entries must be zero ($a_{ii}=-a_{ii}⟹a_{ii}=0$), which is a non-obvious constraint often missed.
    • Study strategy: Always check the diagonal first; if any element is non-zero, it is not skew-symmetric.

• Action Plan

  • Immediate review actions:
    • Review the derivation of the commuting matrix $B$ in Activity 2 to understand the variable constraints ($x=w,z=0$).
    • Verify the proof that $A-A^T$ is skew-symmetric using transpose properties.
  • Practice and application:
    • Answer Remember/Understand questions
    • Answer Apply/Analyze questions
    • Attempt Evaluate/Create questions
  • Deep dive study:
    • Focus on Challenging Concepts (above)
    • Extend atomic notes only where the new lecture adds content
  • Verification and integration:
    • Cross-reference notes with textbook/slides
    • Identify remaining gaps
    • Link this lecture to prior/future lecture notes (lecture notes only; no MOCs)

• Footnotes