Journal

Tag: permutation-matrix

Long-form notes, Qiskit experiments, and applied optimization thinking.

September 1, 2026•Learning Notes•7 min read

Understanding Transposes and Permutations: Why Rows and Columns Change Roles

A practical path through transpose and permutation matrices: why transpose is needed, how it connects rows and columns, why (AB)^T reverses order, why permutation matrices satisfy P^{-1}=P^T, and how left multiplication PA actually rearranges rows.

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August 30, 2026•Learning Notes•6 min read

Why LU Matters for Large Systems: From Ax = b to Pivoting and PA = LU

Gaussian elimination is easy to demonstrate on small matrices, but real systems may contain thousands or millions of unknowns. This note follows the question of how computers solve large Ax = b systems, why LU makes repeated solutions efficient, and why zero or tiny pivots lead naturally to row exchanges and PA = LU.

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