Matrix Multiplication

Here is a concise summary for your Obsidian Vault, using the same two matrices to show how each method builds the final result.

The Setup

$A=\left[\begin{array}{cc}1& 2\\ 3& 4\end{array}\right],B=\left[\begin{array}{cc}5& 6\\ 7& 8\end{array}\right]$ Result $C=\left[\begin{array}{cc}19& 22\\ 43& 50\end{array}\right]$

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1. The “Point” View (Row times Column)

Focus: Computing one specific number ($c_{ij}$) at a time.

• Method: Take the dot product of Row $i$ of $A$ and Column $j$ of $B$.
• Example ($c_{11}$): $(1\times 5)+(2\times 7)=19$.
• Contribution: Every entry is an isolated “inner product” calculation.

2. The “Column” View (Matrix times Column)

Focus: Computing one whole column of $C$ at a time.

• Method: Each column of $C$ is a linear combination of the columns of $A$.
• Example (Column 1 of $C$): $5\left[\begin{array}{c}1\\ 3\end{array}\right]+7\left[\begin{array}{c}2\\ 4\end{array}\right]=\left[\begin{array}{c}19\\ 43\end{array}\right]$.
• Contribution: The columns of $B$ provide the “multipliers” for the columns of $A$.

3. The “Row” View (Row times Matrix)

Focus: Computing one whole row of $C$ at a time.

• Method: Each row of $C$ is a linear combination of the rows of $B$.
• Example (Row 1 of $C$): $1+2=$.
• Contribution: The rows of $A$ provide the “multipliers” for the rows of $B$.

4. The “Layer” View (Columns times Rows)

Focus: Summing full-sized matrices.

• Method: $C$ is the sum of ($A$’s column $k\times B$’s row $k$).
• Example:
  • Layer 1: $\left[\begin{array}{c}1\\ 3\end{array}\right]=\left[\begin{array}{cc}5& 6\\ 15& 18\end{array}\right]$
  • Layer 2: $\left[\begin{array}{c}2\\ 4\end{array}\right]=\left[\begin{array}{cc}14& 16\\ 28& 32\end{array}\right]$
• Contribution: Each rank-one matrix “layer” adds one specific term to every dot product sum in the final result.

5. The “Chunk” View (Block Multiplication)

Focus: Breaking large matrices into smaller sub-matrices.

• Method: Partition $A$ and $B$ into blocks and multiply them as if they were numbers.
• Example: If we treat each entry of $A$ and $B$ as a $1\times 1$ block, the top-left block of $C$ is $(A_{11}{B}_{11})+(A_{12}{B}_{21})$.
• Contribution: It groups individual arithmetic steps into matrix “packets,” which is how high-level computers process data.

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Does this organization work for your vault, or would you like to see how Gaussian Elimination fits into this matrix language next?