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A symmetric matrix is a square matrix that is identical to its transpose. This means if you flip the matrix over its diagonal, it remains unchanged. For a matrix \( A \) to be symmetric, we have the condition \( a_{ij} = a_{ji} \) for all elements. In the context of an inner product \( \langle \mathbf{u}, \mathbf{v} \rangle = \mathbf{u}^T A \mathbf{v} \), using a symmetric matrix ensures the inner product maintains certain mathematical properties, such as symmetry in the arguments. A crucial characteristic of symmetric matrices is that they always produce real eigenvalues. Additionally, they play a significant role in defining quadratic forms and are widely used in optimization problems.
Recognizing a matrix as symmetric is straightforward. Consider the matrix \( A = \begin{bmatrix} 1 & 2 \ 2 & 5 \end{bmatrix} \). Note that swapping the rows and columns results in the same matrix, confirming its symmetry.

Matrix multiplication is a fundamental operation in linear algebra, notable for its non-commutative nature. This means that generally \( AB eq BA \). However, it maintains associative and distributive properties. When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. In our problem, we perform the multiplication as follows:

• The rows of the first matrix (transpose of \( \mathbf{u} \)) by the columns of the second matrix (\( \mathbf{v} \)).
• Each element of the resulting matrix is obtained by taking the dot product of corresponding row and column vectors.

In our problem, matrix multiplication simplifies the expression \( u_1v_1 + 2u_1v_2 + 2u_2v_1 + 5u_2v_2 \) to \( (u_1, u_2) \begin{bmatrix} 1 & 2 \ 2 & 5 \end{bmatrix} (v_1, v_2)^T \). This compact representation facilitates mathematical operations and analysis, especially in computational scenarios.

In the context of determining the symmetric matrix \( A \) for an inner product, coefficients comparison is a vital technique. This involves matching coefficients from an expanded inner product expression with those derived from matrix multiplication. When given \( \langle \mathbf{u}, \mathbf{v} \rangle = u_1v_1 + 2u_1v_2 + 2u_2v_1 + 5u_2v_2 \), we set this equal to the expression obtained from the matrix product \( (u_1, u_2) \begin{bmatrix} a & b \ b & d \end{bmatrix} (v_1, v_2)^T \).
The task is to identify:

• \( a = 1 \) because it is the coefficient of \( u_1v_1 \).
• \( b = 2 \), since it appears in \( 2u_1v_2 \) and \( 2u_2v_1 \).
• \( d = 5 \) as it matches the coefficient of \( u_2v_2 \).

By systematically comparing coefficients, we derive \( A = \begin{bmatrix} 1 & 2 \ 2 & 5 \end{bmatrix} \), ensuring that the algebraic structure aligns with the original expression.