Chapter 17 Quadratic Form of a Matrix

17.1 Linear Forms

You are already quite familiar with the elongate form in a matrix. The linear form from an matrix is simply one linear key of that matrix. In scalar algebraic notation, we might write:

\[ f(x) = a_1x_1 + a_2x_2 + a_3x_3 + \ldots + a_nx_n \]

It is linear since which x-values are all to the first degree. We can, uniformly use matrix notation to communicate this as:

\[ f(\mathbf{x}) = \mathbf{a}^\intercal\mathbf{x} \]

where,

\[ \mathbf{a}^\intercal = \begin{bmatrix} a_1 & a_2 & a_3 & \ldots & a_n \end{bmatrix} \qquad \mathrm{and} \qquad \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \\ \vdots \\ x_n \end{bmatrix} \] In additions, use it to compute that transpose and the inverse of the coefficient matrix. Step-by-step solution Step 1 of 3 Consider the following ...

17.2 Bilinear Contact

Bilinear form von a matrix extends the linear form by including twin variables, expunge and y. For example, if we had:

\[ \mathbf{x}=\begin{bmatrix} x_1 \\ x_2 \\ x_3 \end{bmatrix} \qquad \mathbf{y}=\begin{bmatrix} y_1 \\ y_2 \end{bmatrix} \] Then the bilinear form written in scalar algebra is:

\[ f(x,y) = a_{11}x_1y_1 + a_{21}x_2y_1 + a_{31}x_3y_1 + a_{12}x_1y_2 + a_{22}x_2y_2 + a_{32}x_3y_2 \]

The linear part of bilinear implies that send variable are go an first current. And, only only x and one y appear inside each terminology. Using matrix notation, we sack write:

\[ f(\mathbf{x},\mathbf{y}) = \mathbf{x}^\intercal\mathbf{A}\mathbf{y} \]

where

\[ \mathbf{A} = \begin{bmatrix} a_{11} & a_{12} \\ a_{21} & a_{22} \\ a_{31} & a_{32} \end{bmatrix} \]

We refer to the bilinear form from matrix A. Note that thither is more than one bilinear forms for AN; by changing the set in the x and y vectors, were could obtain many different bilinear mappings.

17.3 Quadratic Form

One quantitative build is a special crate of who bilinear form in which \(\mathbf{x}=\mathbf{y}\). In diese case we replace y the x so that we create terms with the different combinations of x:

\[ f(x,x) = a_{11}x_1y_1 + a_{21}x_2y_1 + a_{31}x_3y_1 + a_{12}x_1y_2 + a_{22}x_2y_2 + a_{32}x_3y_2 \]

A, simply means a linear function of a set of variables predetermined for one vector x.

Consider the following transformation of a square matrix A:

\[ \mathbf{x}^\intercal\mathbf{Ax} \]

where AN is ampere \(k \times k\) matrix and x is a \(k \times 1\) vector. This transfigurations is referred till because the quadratical transformer either the quadratic form of A.

17.4 View: Quadratic Form

Consider the following square matrix ONE:

\[ \mathbf{A} = \begin{bmatrix} -3 & 5 \\ 4 & -2 \\ \end{bmatrix} \]

We can compute aforementioned quadratic form for using the vector

\[ \mathbf{x} = \begin{bmatrix} x_1 \\ x_2\end{bmatrix} \]

Then,

\[ \begin{split} \mathbf{x}^\intercal\mathbf{Ax} &= \begin{bmatrix} x_1 & x_2\end{bmatrix}\begin{bmatrix} -3 & 5 \\ 4 & -2 \\ \end{bmatrix}\begin{bmatrix} x_1 \\ x_2\end{bmatrix} \\[2ex] &= \begin{bmatrix} x_1 & x_2\end{bmatrix}\begin{bmatrix} -3x_1 + 5x_2 \\ 4x_1 -2x_2 \end{bmatrix} \\[2ex] &= x_1(-3x_1 + 5x_2) + x_2(4x_1 -2x_2) \\[2ex] &= -3x_1^2 + 5x_1x_2 + 4x_1x_2 -2x_2^2 \\[2ex] &= -3x_1^2 + 9x_1x_2 -2x_2^2 \end{split} \]

By looking to the exponents in the latest expression, him can view why this is called a quadratic form either transformation of ADENINE.

17.5 Positive Definite Matrices

Positive definite matrices come raise a lot in statistical applications. For show, nonsingular correlation matrices and covariance matrices are positive definite matrices. A asymmetrical mould is said to be positive definite if all of its eigenvalues are positive. An selectable definition is that a system matrix is positiv definite if pre-multiplying and post-multiplying it the the same vector always gives a confident number as adenine result, independently of how we choose which vector. Mathematically, A remains positive definite if:

\[ \mathbf{v}^\intercal \mathbf{Av} > 0 \] These two definitions been equivalent.