Quadratic forms — positive definite, negative definite, indefinite: Foundations

Before you can even read the parent note Quadratic forms — positive definite, negative definite, indefinite, you need to own every symbol it uses. This page builds each one from nothing, in the order they depend on each other.

1. The vector $x$ — an arrow made of numbers

The picture: in 2D, $x = (x_{1} x_{2})$ is an arrow from the origin to the point $(x_{1}, x_{2})$ . Look at the figure: the red arrow is the vector; its two components are how far right and how far up it reaches.

Figure — Quadratic forms — positive definite, negative definite, indefinite

Why the topic needs it: the machine $Q$ eats a vector and gives back a number. Everything below is about what happens to that arrow.

Recall Why write it as a column and not a row?

Because matrix multiplication cares about shape ::: a column $x$ ( $n \times 1$ ) sits neatly to the right of a matrix; the row form $x^{⊤}$ ( $1 \times n$ ) sits to the left.

2. $R^{n}$ — the space the arrow lives in

The picture: $R^{2}$ is the flat plane (every point on a sheet of paper). $R^{3}$ is all of space. Writing $x \in R^{n}$ (read " $x$ in $R^{n}$ ") just says " $x$ is one of those arrows".

Why the topic needs it: the definiteness question is "for all $x \in R^{n}$ , what sign does $Q$ have?" — we must sweep over the entire space of arrows, not one.

3. The matrix $A$ — a grid of numbers

The picture: think of $A$ as a transformation — a rule that takes any arrow and moves it to a new arrow. It stretches, squashes, and rotates space. In the quadratic form it stores the recipe of the machine.

Why the topic needs it: the entire quadratic form is packaged in $A$ . Diagonal entries carry the squared-term coefficients; off-diagonals carry the cross terms.

4. The transpose $A^{⊤}$ and $x^{⊤}$ — flip along the diagonal

The picture: hold the diagonal fixed and reflect everything else through it — like folding paper along the top-left-to-bottom-right crease. The figure shows a matrix and its mirror image.

Why the topic needs it: the form is written $x^{⊤} A x$ — the leading $x^{⊤}$ is a row, and "symmetric" ( $A^{⊤} = A$ ) is the key property the parent demands.

5. Symmetric — the mirror leaves $A$ unchanged

The picture: a symmetric matrix looks like a Rorschach blot across the diagonal. In $(3221)$ the two $2$ 's are mirror twins.

Why the topic needs it: only the symmetric part of a matrix affects $x^{⊤} A x$ , so we always choose $A$ symmetric. Symmetric matrices also have a beautiful guarantee (real eigenvalues, perpendicular eigenvectors) — see Symmetric Matrices and the Spectral Theorem.

6. Matrix–vector multiplication and $x^{⊤} A x$

For $2 \times 2$ this expands cleanly. With $A = (a b b c)$ : $x^{⊤} A x = a x_{1}^{2} + 2 b x_{1} x_{2} + c x_{2}^{2} .$

Why the topic needs it: this is the quadratic form. Every result in the parent is a statement about the sign of this one number.

7. Scalar — one number, and why it equals its own transpose

Why the topic needs it: the parent uses "a scalar equals its own transpose" to kill the antisymmetric part of a matrix. That trick only works because $x^{⊤} A x$ is a scalar.

8. Degree-2 / homogeneous — every term is a product of exactly two coordinates

The picture: if you double the input arrow, the output number quadruples ( $Q (2 x) = 4 Q (x)$ ) — a hallmark of pure degree-2. Figure below shows the parabola-like scaling.

Why the topic needs it: because there are no linear or constant terms, $Q (0) = 0$ always. So the whole classification question only makes sense for $x \neq = 0$ — that is why every definition says "for all $x \neq = 0$ ".

9. Eigenvalues $λ$ and eigenvectors — directions $A$ only stretches

The picture: most arrows swing to a new direction under $A$ ; an eigenvector stays on its own line. The red arrow in the figure lands on the same line, just rescaled by $λ$ .

Why the topic needs it: the master result is that in eigenvector coordinates the machine becomes $Q = λ_{1} y_{1}^{2} + λ_{2} y_{2}^{2} + \dots$ — pure squares. Then the signs of the $λ_{i}$ decide everything. Full details live in Eigenvalues and Eigenvectors.

x \in R^{n}

mean in plain words?

x

is an arrow made of

n

real-number components, living in

n

-dimensional space.

What does the transpose symbol

^{⊤}

do?

Flips a matrix across its main diagonal (rows become columns); turns a column vector into a row.

What makes a matrix symmetric?

A^{⊤} = A

, i.e.

A_{ij} = A_{j i}

— mirror twins across the diagonal.

Why is

x^{⊤} A x

a single number, not a matrix?

Row times matrix times column collapses (

1 \times n

)(

n \times n

)(

n \times 1

) down to a

1 \times 1

scalar.

Expand

x^{⊤} A x

for

A = (a b b c)

a x_{1}^{2} + 2 b x_{1} x_{2} + c x_{2}^{2}

What is an eigenvector, in one sentence?

An arrow

v

that

A

only stretches (by factor

λ

) without rotating:

A v = λ v

What does

λ = 0

signify geometrically?

A direction

A

collapses to zero; it makes

det A = 0

and marks the semidefinite borderline.

What does an orthogonal matrix

Q

do to space?

Rotates (or reflects) it, preserving all lengths and angles;

Q^{⊤} Q = I

What is

D_{2}

for

A = (3221)

det = 3 (1) - 2 (2) = - 1

Why must definiteness be tested only for

x \neq = 0

Because a degree-2 form always gives

Q (0) = 0

, so the zero vector says nothing about sign.

Foundations — Quadratic forms — positive definite, negative definite, indefinite

1. The vector $x$ — an arrow made of numbers

2. $R^{n}$ — the space the arrow lives in

3. The matrix $A$ — a grid of numbers

4. The transpose $A^{⊤}$ and $x^{⊤}$ — flip along the diagonal

5. Symmetric — the mirror leaves $A$ unchanged

6. Matrix–vector multiplication and $x^{⊤} A x$

7. Scalar — one number, and why it equals its own transpose

8. Degree-2 / homogeneous — every term is a product of exactly two coordinates

9. Eigenvalues $λ$ and eigenvectors — directions $A$ only stretches

10. $Λ$ , $diag$ , and orthogonal $Q$

11. Determinant $det$ and leading principal minors $D_{k}$

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