What does $x^TAx$?

What does the following mean? I know that $x^Tx$ is the magnitude of $x$. What does the following formula represent intuitively? $x$ is a vector and $A$ is some scaling matrix. The given is $x^T A x i$

(Ignore the i)

• Mar 16, 2018 at 1:20
• Note that $x$ and $X$ are two different things. Please do not pretend they are the same. Mar 16, 2018 at 1:28
• Wikipedia: Definite quadratic form will be helpful. $x^TAx$ comes from a matrix form of a quadratic form.
– ChoF
Mar 16, 2018 at 2:41
• $x^Tx$ is the square of the magnitude of $x$ (assuming, that is, that the coordinates are relative to the standard basis).
– amd
Mar 17, 2018 at 3:31

Rather than thinking about $y^T x$, it's better to think about the dot product $y \cdot x$. Recall that in a real vector space, the dot product $y \cdot x$ gives you a measure of how the vectors $x$ and $y$ project onto each other: if the dot product is large and positive, the vectors $x$ and $y$ are pointing in a similar direction, and if it is large and negative, the vectors are pointing in an opposite direction.

Since $x^T A x = x \cdot (Ax)$, it is a measure of how "pointing in the same direction" the vector $Ax$ is compared to $x$. The matrix $A$ is called positive definite if $x \cdot (Ax) > 0$ for all $x \neq 0$. This gives some immediate consequences:

• $A$ has full rank, so is an invertible matrix.
• For each vector $x$, the vectors $x$ and $Ax$ are always on the same side of the hyperplane perpendicular to $x$.
• Thank you for this insight. This cleared up a few things for me! Jan 25, 2021 at 19:55
• The dot approach really helps. Great answer. Jun 22, 2021 at 15:19

If this was in one dimension, what would you get?

You would get a second degree polynomial $$ax^2$$. Imagine now instead dimension $$37$$ and a function $$f(x_1,x_2 ... , x_{37})=x_1^2 + x_2^2 + ... + x_{37}^2$$. This is very cumbersome to type, $$x^TAx$$ is a generalization to higher dimension, a much more compact form to write a second degree polynomial instead of writing all terms out.

If $X$ is a vector in $n$ dimensional space, it is an $n \times 1$ matrix-a column. $X^T$ is a $1 \times n$ matrix-a row. $A$ is an $n \times n$ matrix, which you might see as the matrix of a linear transformation on your space. $AX$ is the vector that $A$ transforms $X$ into. It is again $n \times 1$. Then $X^TAX$ is a $1 \times 1$ matrix, a number. $A$ is positive definite if the number is positive for all nonzero $X$. An example is the identity matrix, because $IX=X$ and we know $X^TX$ is positive for all nonzero $X$.