# Notes - Linear Algebra II HT23, Quadratic forms

### Flashcards

A quadratic form in $n$ variables can be written as

\[Q(x_1, \ldots, x_n) = \sum_{i, j = 1}^n a_{ij} x_i x_j\]
where $a _ {ij}$ represents the coefficients of each term. How could you write this in terms of matrix multiplication?

where

\[x = \begin{pmatrix}x_1 \\\\ \vdots \\\\ x_n\end{pmatrix}\]A quadratic form in $n$ variables over $\mathbb R$ can be written as

\[Q(x_1, \ldots, x_n) = \sum_{i, j = 1}^n a_{ij} x_i x_j = x^\intercal \mathbf A x\]
What can be made true about the matrix $\mathbf A$?

It is symmetric.

Since any quadratic form in $n$ variables over $\mathbb R$ can be written as

\[Q(x_1, \ldots, x_n) = x^\intercal \mathbf A x\]
where $\mathbf A$ is real and symmetric, what useful consequence comes about due to the spectral theorem for symmetric matrices?

There exists a change of orthogonal change of variables so that

\[Q(y_1, \ldots , y_n) = y^\intercal \mathbf \Lambda y = \lambda_1 y_1^2 + \cdots + \lambda_n y_n^2\]where $\mathbf \Lambda$ is a diagonal matrix and $\lambda _ 1, \ldots, \lambda _ n$ are eigenvalues.

What is a quadric?

A set of points in $\mathbb R^3$ satisfying

\[x^\intercal \mathbf A x + \mathbf b x + \pmb c = 0\]What is a conic?

A set of points in $\mathbb R^2$ satisfying

\[x^\intercal \mathbf A x + \mathbf b x + \pmb c = 0\]A quadric is a set of points in $\mathbb R^3$ satisfying

\[x^\intercal \mathbf A x + \mathbf b x + \pmb c = 0\]
due to the spectral theorem, and some completing the square magic, there’s actually a much simpler way of classifying quadrics under an orthogonal change of variables $Y _ 1, \ldots Y _ 3$. What are the only possibilities?

What upper and lower bounds can you put on the quadratic form

\[x^\intercal \mathbf A x\]
where $A$ is a symmetric matrix?

where $\lambda _ 1$ is the smallest eigenvalue, $\lambda _ n$ is the largest.

For any quadratic form we have

\[\lambda_1 x^\intercal x \le x^\intercal \mathbf A x \le \lambda_n x^\intercal x\]
where $\lambda _ 1$ is the smallest eigenvalue, $\lambda _ n$ is the largest. When are the bounds achieved?

When $x$ is in the eigenspace of $\lambda _ 1$ or $\lambda _ n$.