91Ó°ÊÓ

Matrix multiplication is a fundamental concept in linear algebra. It involves the multiplication of two matrices to produce a third matrix. In this exercise, we have two matrices: a vector transpose, \( \mathbf{x}^T \), and a matrix, \( A \). The goal is to multiply them together to achieve a new matrix.

• To perform matrix multiplication, align the rows of the first matrix with the columns of the second matrix.
• Calculate the sum of the products of the corresponding elements from the row of the first matrix and the column of the second matrix.
• The resulting product matrix will have dimensions based on the number of rows of the first matrix and columns of the second matrix.

The expression \( \mathbf{x}^T A \) is achieved by multiplying the transpose of the vector by the matrix \( A \), carefully summing the appropriate products to form a new row vector.

The transpose of a vector is a straightforward yet crucial mathematical operation. It essentially involves flipping a column vector into a row vector, or vice versa.

In the exercise, we start with a column vector \( \mathbf{x} = \begin{bmatrix} x \ y \ z \end{bmatrix} \). Its transpose, denoted as \( \mathbf{x}^T \), alters its shape to \( \begin{bmatrix} x & y & z \end{bmatrix} \). This change transforms it, making it compatible for matrix multiplication with \( A \).

The transpose is crucial for quadratic forms like \( f(\mathbf{x}) = \mathbf{x}^T A \mathbf{x} \), ensuring the dimensions align correctly for subsequent operations.

A matrix-vector product is the process of multiplying a matrix by a vector, producing a new vector. It is a key operation used in numerous applications, such as solving systems of equations and transforming geometrical objects.

In the quadratic form \( f(\mathbf{x}) \), after computing \( \mathbf{x}^T A \), we yield a vector. This new vector is then multiplied by the original vector \( \mathbf{x} \), resulting in a scalar output.

• Each element in the new vector is obtained by performing a dot product between a row of the matrix and the vector.
• The resulting vector's length corresponds to the number of rows in the matrix.
• This scalar output reflects the energy or magnitude associated with the quadratic form.

The final expression, after simplifying these products, gives us the canonical form of the quadratic function: \( f(\mathbf{x}) = x^2 + 2y^2 + 3z^2 - 6xz + 2yz \). This succinctly encapsulates the combined effects of all the elements in both the matrix and vector.