91Ó°ÊÓ

Matrix-vector multiplication is a fundamental operation in linear algebra that is found in many mathematical problems. It is essential when evaluating expressions like quadratic forms. A matrix-vector multiplication involves combining a matrix and a vector to produce a new vector.
For a matrix \[A = \begin{bmatrix} a_{11} & a_{12} & \dots & a_{1n} \ a_{21} & a_{22} & \dots & a_{2n} \ \vdots & \vdots & \ddots & \vdots \ a_{m1} & a_{m2} & \dots & a_{mn} \end{bmatrix}\]and a vector
\[\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ \vdots \ x_n \end{bmatrix},\]
the multiplication \(A\mathbf{x}\) results in a new vector \(\mathbf{b}\), where each element \(b_i\) is computed as \(b_i = a_{i1}x_1 + a_{i2}x_2 + \dots + a_{in}x_n\).

• Each row of the matrix corresponds to an element in the resulting vector after multiplication.
• The operation involves summing the products of corresponding elements in the row of the matrix and the column vector.

In our exercise, given an example matrix \(A\) and vector \(\mathbf{x}\), we computed the product \(A\mathbf{x}\) to obtain a new vector \(\begin{bmatrix} 6 \ 5 \ 5 \end{bmatrix}\). This computation is central to solving the quadratic form.

A symmetric matrix has a special property where the matrix is equal to its transpose. This means, for a matrix \(A\), it is symmetric if \(A = A^T\). Symmetric matrices arise frequently in mathematics, especially when discussing quadratic forms.
A few interesting properties of symmetric matrices include:

• Only square matrices (same number of rows and columns) can be symmetric.
• The entries on the main diagonal can take any value, but for symmetry, the entries across the diagonal must be equal.

In our context, the matrix \(A\) is symmetric, meaning that its role in the quadratic form ensures certain computational properties. This symmetry simplifies calculations because when calculating terms like \(\mathbf{x}^{T} A \mathbf{x}\), we do not have to worry about differences when exchanging rows and columns. Understanding the symmetry can make it easier to work with such matrices efficiently in various mathematical applications.

The transpose of a matrix is a new matrix achieved by exchanging the rows and columns of the original matrix. This operation is denoted by the superscript \(T\). If we have a matrix \(A\), its transpose \(A^T\) is formed by turning rows into columns and columns into rows.
For example, if \(A\) is a matrix \(\begin{bmatrix} a_{11} & a_{12} \ a_{21} & a_{22} \end{bmatrix}\), its transpose \(A^T\) would be \(\begin{bmatrix} a_{11} & a_{21} \ a_{12} & a_{22} \end{bmatrix}\).

• Transposing a matrix twice returns the original matrix: \((A^T)^T = A\).
• The dot product in a quadratic form, \(\mathbf{x}^T A \mathbf{x}\), involves vector and matrix transpositions in the steps.

In our exercise, \(\mathbf{x}^T = \begin{bmatrix} 1 & 2 & 3 \end{bmatrix}\) is the transpose of the column vector \(\mathbf{x}\). The calculation involving matrix transpose is important for assessing how a matrix interacts with a vector or another matrix. It helps to understand how orientation and dimension change under transposition.