Comprehensive Study Notes on Matrix Operations and Transposition

Course Operations and Exam Guidance

• Test 1 Review: Test 1 review is scheduled for Wednesday. Attendance is mandatory.
• Lecture Note Availability: Board notes are not posted online. In-person class attendance is essential for capturing all course material.
• Key Rotation Problems: Rotation problems—specifically those combining two sequential process rotations—are critical exam topics. Standard textbook problems do not provide combined rotation examples outside of reference problems.
• Exam Preparation Strategy: Test 1, Midterm 2, and Final Exam problems directly mirror the homework and reference problems. The difficulty level of exam questions will not exceed that of homework problems or lecture examples. Practicing reference and homework problems until full familiarity is achieved is required.

Methods of Matrix Multiplication

Matrix multiplication can be computed using three distinct approaches: column-wise multiplication, entry-wise dot products, and row-wise multiplication.

• Definition 1: Column-Wise Method (Matrix-Vector Multiplication)

  • The matrix product $AB$ can be computed column by column as linear combinations of the columns of $A$.
  • Let $\text{Column}_j(B) = \bar{b}_j$ represent the $j$-th column of matrix $B$.
  • The $j$-th column of the product $AB$ is given by:         $\text{Column}_j(AB) = A \bar{b}_j$
  • Specifically, the first column of $AB$ is $A \bar{b}_1$, and the second column is $A \bar{b}_2$.

• Definition 2: Entry-Wise Method (Row-Column Product)

  • Let $A(i, :)$ denote the $i$-th row of matrix $A$ spanning columns $1$ through $n$.
  • The entry in row $i$ and column $j$ of $AB$ is the dot product of the $i$-th row of $A$ and the $j$-th column of $B$:         $(AB)_{ij} = \sum_{k=1}^{n} a_{ik} b_{kj}$

• Definition 3: Row-Wise Method (Row-Matrix Multiplication)

  • The $i$-th row of the product $AB$ is obtained by multiplying the $i$-th row vector of $A$ by the entire matrix $B$:         $\text{Row}_i(AB) = \text{Row}_i(A) B$
  • Multiplying a $1 \times n$ row vector by an $n \times p$ matrix $B$ yields a $1 \times p$ row vector.

• Numerical Demonstration of Row-Wise Computation:

  • Given matrix $A$ with dimensions $2 \times 3$:         $A = \begin{pmatrix} 4 & 3 & 6 \\ 1 & -2 & 3 \end{pmatrix}$
  • First row calculation of $AB$:         $\text{Row}_1(AB) = \text{Row}_1(A) B = \begin{pmatrix} 4 & 3 & 6 \end{pmatrix} B = \begin{pmatrix} 11 & 0 & 21 \end{pmatrix}$
  • Second row calculation of $AB$:         $\text{Row}_2(AB) = \text{Row}_2(A) B = \begin{pmatrix} 1 & -2 & 3 \end{pmatrix} B$
    • Entry $(2,1)$: $1(4) - 5 = -1$
    • Entry $(2,2)$: $3 + 10 = 11$
    • Entry $(2,3)$: $6 - \dots$

Algebraic Properties of Matrix Multiplication

Let $A$, $B$, and $C$ be matrices with dimensions such that the indicated sums and products are defined.

• Associative Law of Multiplication:     $(AB)C = A(BC)$
• Left Distributive Law:     $A(B + C) = AB + AC$
• Right Distributive Law:     $(B + C)A = BA + CA$
• Identity Matrix Multiplication:
  • For an $m \times n$ matrix $A$:         $I_m A = A$$A I_n = A$
  • $I_m$ denotes the $m \times m$ identity matrix, and $I_n$ denotes the $n \times n$ identity matrix.
  • If $A$ is non-square ($m \neq n$), the left and right identity matrices must match the respective outer dimension. If $A$ is itself an identity matrix, then $I_m = A = I_n$.

Non-Commutativity and Non-Cancellation Laws

• Non-Commutativity ($AB \neq BA$):

  • In general scalar arithmetic, $5 \times 7 = 7 \times 5$, but matrix multiplication does not commute in general: $AB \neq BA$
  • Explicit Counterexample:         Let $A = \begin{pmatrix} 5 & 1 \\ 3 & -2 \end{pmatrix}$ and $B = \begin{pmatrix} 2 & 0 \\ 4 & 3 \end{pmatrix}$
  • Compute $AB$ via row-column multiplication:         $(AB)_{11} = 5(2) + 1(4) = 10 + 4 = 14$$(AB)_{12} = 5(0) + 1(3) = 3$$(AB)_{21} = 3(2) + (-2)(4) = 6 - 8 = -2$$(AB)_{22} = 3(0) + (-2)(3) = -6$$AB = \begin{pmatrix} 14 & 3 \\ -2 & -6 \end{pmatrix}$
  • Compute $BA$ via row-column multiplication:         $(BA)_{11} = 2(5) + 0(3) = 10$$(BA)_{12} = 2(1) + 0(-2) = 2$$(BA)_{21} = 4(5) + 3(3) = 20 + 9 = 29$$(BA)_{22} = 4(1) + 3(-2) = 4 - 6 = -2$$BA = \begin{pmatrix} 10 & 2 \\ 29 & -2 \end{pmatrix}$
  • Comparison reveals $AB \neq BA$. Thus, matrices $A$ and $B$ do not commute.

• Failure of Matrix Cancellation:

  • The equation $AB = AC$ is logically equivalent to $A(B - C) = \mathbf{0}$. Setting $D = B - C$, this becomes $AD = \mathbf{0}$.
  • If $A \neq \mathbf{0}$, it does not guarantee that $D = \mathbf{0}$ (i.e., $B = C$).
  • Similarly, for matrix-vector systems $A \bar{x} = \mathbf{0}$ with $A \neq \mathbf{0}$, $\bar{x}$ is not required to be $\mathbf{0}$.
  • Linear Dependence Criterion: If the columns of matrix $A$ are linearly dependent, then $A \bar{x} = \mathbf{0}$ possesses non-zero (nontrivial) solutions.
  • Example: For $A = \begin{pmatrix} 1 & 2 & 3 \\ 4 & 5 & 6 \end{pmatrix}$, the three column vectors are linearly dependent, ensuring non-zero solution vectors $\bar{x} = \begin{pmatrix} x_1 \\ x_2 \\ x_3 \end{pmatrix}$.
  • Key Takeaways:
    1. $AB = AC$ does not imply $B = C$ in general.
    2. $AB = \mathbf{0}$ does not imply $A = \mathbf{0}$ or $B = \mathbf{0}$.

Powers and Transposition of Matrices

• Powers of a Matrix:

  • Matrix powers $A^k$ are defined only for square matrices ($n \times n$).
  • For a positive integer $k$:         $A^k = \underbrace{A \cdot A \cdots A}_{k \text{ times}}$

• Transpose of a Matrix:

  • If $A$ is an $m \times n$ matrix, its transpose $A^T$ is an $n \times m$ matrix.
  • The columns of $A^T$ are constructed directly from the rows of $A$.
  • Example:         For a $3 \times 2$ matrix $A = \begin{pmatrix} a & b \\ c & d \\ e & f \end{pmatrix}$, its $2 \times 3$ transpose is:         $A^T = \begin{pmatrix} a & c & e \\ b & d & f \end{pmatrix}$

• Properties of Matrix Transposition:     Let $A$ and $B$ be matrices of appropriate dimensions, and let $r$ be a scalar:

  1. $(A^T)^T = A$
  2. $(A + B)^T = A^T + B^T$
  3. $(rA)^T = r A^T$
  4. $(AB)^T = B^T A^T$

Proof: Transpose of a Matrix Product

• Theorem: $(AB)^T = B^T A^T$
• Proof Procedure:
  1. Consider the $j$-th column of $(AB)^T$.
  2. By the definition of transposition, the $j$-th column of $(AB)^T$ equals the transpose of the $j$-th row of $AB$.
  3. By row-matrix multiplication, the $j$-th row of $AB$ is given by:        $\text{Row}_j(AB) = \text{Row}_j(A) B$
  4. Expressing $B$ in terms of its columns $\begin{pmatrix} \bar{b}_1 & \bar{b}_2 & \dots & \bar{b}_n \end{pmatrix}$, the $j$-th row vector is:        $\begin{pmatrix} \text{Row}_j(A) \bar{b}_1 & \text{Row}_j(A) \bar{b}_2 & \dots & \text{Row}_j(A) \bar{b}_n \end{pmatrix}$
  5. Transposing this row vector yields the $j$-th column vector of $(AB)^T$:        $\begin{pmatrix} \text{Row}_j(A) \bar{b}_1 \\ \text{Row}_j(A) \bar{b}_2 \\ \vdots \\ \text{Row}_j(A) \bar{b}_n \end{pmatrix}$
  6. Now examine the $j$-th column of $B^T A^T$. By definition, the $j$-th column of $B^T A^T$ is $B^T$ multiplied by the $j$-th column of $A^T$.
  7. The $j$-th column of $A^T$ is $\left(\text{Row}_j(A)\right)^T$, and the rows of $B^T$ are $\bar{b}_1^T, \bar{b}_2^T, \dots, \bar{b}_n^T$.
  8. Matrix multiplication yields the identical column vector, completing the proof: $(AB)^T = B^T A^T$.

In-Class Assessment Protocol

• Duration: 10 minutes.
• Rules: Strict silence; no discussion or collaboration allowed. Unauthorized communication results in an automatic score of zero.
• Format: Memory-based evaluation with binary response format ("Yes" for true statements, "No" for false statements).
• Submission: Write full name at the top of the paper, record answers, and hand in at the front upon completion.