Question

By direct matrix multiplication, express each of the following matrix products as a quadratic form: \(\left(\begin{array}{lll}a & {\left[\begin{array}{ll}u & v\end{array}\right]\left[\begin{array}{ll}4 & 2 \\ 2 & 3\end{array}\right]\left[\begin{array}{l}u \\\ v\end{array}\right]}\end{array}\right]\) \((b)\left[\begin{array}{ll}u & v\end{array}\right]\left[\begin{array}{rr}-2 & 3 \\ 1 & -4\end{array}\right]\left[\begin{array}{l}u \\ v\end{array}\right]\) \((c)\left[\begin{array}{ll}x & y\end{array}\right]\left[\begin{array}{ll}5 & 2 \\\ 4 & 0\end{array}\right]\left[\begin{array}{l}x \\ y\end{array}\right]\) \((d)[d x \quad d y]\left[\begin{array}{ll}f_{x x} & f_{x y} \\ f_{y x} & f_{y y}\end{array}\right]\left[\begin{array}{l}d x \\ d y\end{array}\right]\)

Short answer

(a) \(4u^2 + 4uv + 3v^2\), (b) \(-2u^2 + 4uv - 4v^2\), (c) \(5x^2 + 6xy\), (d) \(f_{xx}dx^2 + (f_{xy} + f_{yx})dxdy + f_{yy}dy^2\)."

Step-by-step solution

Matrix Multiplication for (a)

For the first expression, we have the vector \([u \ v]\) and the matrix \(\begin{bmatrix}4 & 2 \ 2 & 3\end{bmatrix}\). First, multiply the matrix with the vector on the right:\[\begin{bmatrix}4 & 2 \ 2 & 3\end{bmatrix}\begin{bmatrix}u \ v\end{bmatrix} = \begin{bmatrix}4u + 2v \ 2u + 3v\end{bmatrix}\]Then, multiply the resulting vector with the vector \(\begin{bmatrix}u & v\end{bmatrix}\) on the left:\[\begin{bmatrix}u & v\end{bmatrix}\begin{bmatrix}4u + 2v \ 2u + 3v\end{bmatrix} = u(4u + 2v) + v(2u + 3v) = 4u^2 + 2uv + 2uv + 3v^2 = 4u^2 + 4uv + 3v^2\]So, the quadratic form is \(4u^2 + 4uv + 3v^2\).

Matrix Multiplication for (b)

For the second expression, start by multiplying the matrix \(\begin{bmatrix}-2 & 3 \ 1 & -4\end{bmatrix}\) with the vector \([u \ v]\):\[\begin{bmatrix}-2 & 3 \ 1 & -4\end{bmatrix}\begin{bmatrix}u \ v\end{bmatrix} = \begin{bmatrix}-2u + 3v \ u - 4v\end{bmatrix}\]Then, multiply the result by the vector \(\begin{bmatrix}u & v\end{bmatrix}\) on the left:\[\begin{bmatrix}u & v\end{bmatrix}\begin{bmatrix}-2u + 3v \ u - 4v\end{bmatrix} = u(-2u + 3v) + v(u - 4v) = -2u^2 + 3uv + uv - 4v^2 = -2u^2 + 4uv - 4v^2\]Thus, the quadratic form is \(-2u^2 + 4uv - 4v^2\).

Matrix Multiplication for (c)

For the third expression, we begin by multiplying the matrix \(\begin{bmatrix}5 & 2 \ 4 & 0\end{bmatrix}\) with the vector \([x \ y]\):\[\begin{bmatrix}5 & 2 \ 4 & 0\end{bmatrix}\begin{bmatrix}x \ y\end{bmatrix} = \begin{bmatrix}5x + 2y \ 4x\end{bmatrix}\]Next, multiply the result by the vector \(\begin{bmatrix}x & y\end{bmatrix}\) on the left:\[\begin{bmatrix}x & y\end{bmatrix}\begin{bmatrix}5x + 2y \ 4x\end{bmatrix} = x(5x + 2y) + y(4x) = 5x^2 + 2xy + 4xy = 5x^2 + 6xy\]So, the quadratic form is \(5x^2 + 6xy\).

Matrix Multiplication for (d)

For the fourth expression, first multiply the matrix \(\begin{bmatrix}f_{xx} & f_{xy} \ f_{yx} & f_{yy}\end{bmatrix}\) with the vector \([dx \ dy]\):\[\begin{bmatrix}f_{xx} & f_{xy} \ f_{yx} & f_{yy}\end{bmatrix}\begin{bmatrix}dx \ dy\end{bmatrix} = \begin{bmatrix}f_{xx}dx + f_{xy}dy \ f_{yx}dx + f_{yy}dy\end{bmatrix}\]Then multiply this vector by \(\begin{bmatrix}dx & dy\end{bmatrix}\) on the left:\[\begin{bmatrix}dx & dy\end{bmatrix}\begin{bmatrix}f_{xx}dx + f_{xy}dy \ f_{yx}dx + f_{yy}dy\end{bmatrix} = dx(f_{xx}dx + f_{xy}dy) + dy(f_{yx}dx + f_{yy}dy)\]This simplifies to: \(f_{xx}dx^2 + (f_{xy} + f_{yx})dxdy + f_{yy}dy^2\).Thus, the quadratic form is \(f_{xx}dx^2 + (f_{xy} + f_{yx})dxdy + f_{yy}dy^2\).

Key concepts

Quadratic Form

A quadratic form is a special type of expression where variables are squared and multiplied together. It arises when you have an expression with terms that are linear combinations of squares and products of variables. In the context of matrices, a quadratic form can be represented by point-wise multiplications that include a vector, a matrix, and the transpose of the vector.

To form a quadratic expression from matrices, often you will start with a square matrix and a symmetric product of vectors. For example, the expression \( \left( \begin{bmatrix} u & v \end{bmatrix} \begin{bmatrix} 4 & 2 \ 2 & 3 \end{bmatrix} \begin{bmatrix} u \ v \end{bmatrix} \right) \) results in \(4u^2 + 4uv + 3v^2\).

• The term \(4u^2\) comes from the fact that the \(u\) component is multiplied against itself considering the matrix element \(4\), creating a 'squared' term.
• The product \(4uv\) derives from the cross-multiplication between different components, reflecting the 'interaction' terms between \(u\) and \(v\).
• The term \(3v^2\) is similar to \(4u^2\), but applies to \(v\) due to the matrix diagonal element regarding \(v\).

Understanding quadratic forms is essential in optimization and modeling physical phenomena, where they often describe how one variable affects another in a system.

Matrix Multiplication

Matrix multiplication is a fundamental operation in linear algebra that combines rows and columns of matrices to produce a new matrix. It's vital in expressing various transformations and operations within a mathematical structure.

The multiplication of two matrices requires that the number of columns in the first matrix equals the number of rows in the second. In practical terms:

• Multiply elements in the row of the first matrix by corresponding elements in the column of the second matrix.
• Sum these products to fill each position in the resulting matrix.

When dealing with vectors and matrices, like turning \( \begin{bmatrix} a & b \end{bmatrix} \begin{bmatrix} x \ y \end{bmatrix} \) into a numeric result, each component of the first vector is multiplied by the corresponding component of the matrix. The results are summed to yield a scalar product. This forms the basis of quadratic forms, where multiplication combines both vectors and matrices in a symmetrical way. Such operations are routinely used in physics, computer graphics, and machine learning for data transformation and representation.

Linear Algebra

Linear algebra is a branch of mathematics dealing with vectors, vector spaces, linear transformations, and systems of linear equations. At its core, it seeks to understand mathematical systems with linear properties, which simplifies complex scenarios through structures and straightforward calculations.

Within linear algebra, matrices serve as powerful tools for representing linear transformations and manipulating equations. Learning how vectors interact in this space is crucial:

• Vectors are entities with both direction and magnitude. They are often represented in column or row format.
• Operations involving vectors, like additions or scalar multiplications, maintain linearity, which simplifies solutions.

Matrix operations, such as those seen in multiplying to produce quadratic forms, highlight the utility of these tools. Mastery of linear algebra concepts allows one to solve complex problems in fields from engineering to economics. It forms the backbone of algorithms in computer simulations and optimizations in modern technology and scientific calculations.