Foundations — scipy.linalg — more stable than numpy.linalg, lu, qr, schur

0. What is a vector? (the arrow)

Before matrices we need the thing they act on.

Look at the figure: the arrow's tip lands at the point $(3,2)$. The horizontal component ($3$) is how far right, the vertical component ($2$) is how far up. That's all a vector is — an instruction to move.

Why the topic needs it: everything scipy.linalg does is solve for, transform, or analyse these arrows. $Ax=b$ literally asks "which arrow $x$, after the machine $A$, becomes the arrow $b$?"

1. What is a matrix? (the machine)

The core operation is matrix times vector. Here is the rule, in plain words:

In the figure the blue arrow is the input $\mathbf{v}$; after the machine $A$ acts, it becomes the orange arrow $A\mathbf{v}$ — rotated and stretched. That picture — "a machine that twists and stretches arrows" — is the mental model the whole parent note leans on.

2. The problem $Ax=b$ (the reason all this exists)

Why the topic needs it: LU factorization (the parent's first tool) exists entirely to solve this efficiently and safely.

3. Special shapes of machines

The whole factorization game is about breaking a messy $A$ into machines with easy shapes. Here are the shapes you must recognise on sight.

4. Transpose, inverse, and "orthogonal"

The figure shows why $Q^{T}Q=I$ means "columns are perpendicular unit arrows": entry $(i,j)$ of $Q^{T}Q$ is the dot product of column $i$ with column $j$. On the diagonal that's a column dotted with itself = its length² = $1$; off the diagonal it's two different columns dotted = $0$ because they are perpendicular. (Dot product is next.)

5. Length and the dot product (why "$\Vert \cdot \Vert$" appears)

Why the topic needs it: least squares means "make the leftover arrow $Ax-b$ as short as possible" — that's $\min \Vert Ax-b\Vert$. And "$Q$ preserves length" ($\Vert Q\mathbf{v}\Vert =\Vert \mathbf{v}\Vert$) is the single fact that makes QR-least-squares work.

6. Eigenvalues and eigenvectors (arrows that keep their direction)

Most arrows get twisted (grey, they change direction). The teal arrow is an eigenvector: it comes out pointing the same way, only longer — stretched by $\lambda$. Finding these is what eig and Schur do.

7. Condition number (how nervous is this machine?)

Why the topic needs it: the parent's key stability argument is "normal equations square the condition number, $\kappa (A^{T}A)=\kappa (A{)}^2$" — a machine that was mildly nervous becomes wildly nervous. QR avoids that. See Condition Number.

8. Permutation matrix $P$ (bookkeeping for row swaps)

Why the topic needs it: LU with partial pivoting swaps rows to always divide by the largest available entry (never a tiny one). $P$ remembers which swaps happened, giving $PA=LU$.

The prerequisite map

Read it top-down: arrows and machines feed everything; dot product feeds length and orthogonality; those feed the three factorizations the parent teaches; condition number explains why they're the stable choice.

Related vault topics

• numpy.linalg — the convenience library the parent compares against.
• Gaussian Elimination — the hand process LU records.
• Gram-Schmidt Orthogonalization — how QR's $Q$ is built.
• Eigenvalues and Eigenvectors — deeper on §6.
• Condition Number, Least Squares, Cholesky Decomposition, LAPACK and BLAS.

Equipment checklist