Question

Show that the product \(A A^{\mathrm{T}}\) is a symmetric matrix.

Short answer

The product \(A A^{\text{T}}\) is symmetric because \((A A^{\text{T}})^{\text{T}} = A A^{\text{T}}\).

Step-by-step solution

Understand the Problem

The task involves proving that the product of a matrix and its transpose, denoted as \(A A^{\text{T}}\), is a symmetric matrix. A symmetric matrix is one that is equal to its transpose, so we need to show that \((A A^{\text{T}})^{\text{T}} = A A^{\text{T}}\).

Write the Definition of the Transpose

Recall that the transpose of a matrix \(B\), denoted by \(B^{\text{T}}\), is formed by swapping its rows and columns. Hence for any matrix \(B\), \((B^{\text{T}})_{ij} = B_{ji}\).

Transpose of a Product of Matrices

Use the property of transposes that states \((B C)^{\text{T}} = C^{\text{T}} B^{\text{T}}\) for any matrices \(B\) and \(C\) that can be multiplied together. In this case, \(B = A\) and \(C = A^{\text{T}}\). Thus, we have \(\(\left( A A^{\text{T}} \right)^{\text{T}} = \left( A^{\text{T}} \right)^{\text{T}} A^{\text{T}} = A A^{\text{T}} \).\)

Conclusion

We showed that \((A A^{\text{T}})^{\text{T}} = A A^{\text{T}}\). By definition, this means \(A A^{\text{T}}\) is a symmetric matrix, completing the proof.

Key concepts

Matrix Transpose

In linear algebra, the transpose of a matrix is an important concept. To transpose a matrix, you switch its rows and columns. For instance, if you have a matrix \(A\), its transpose is denoted as \(A^{\text{T}}\). Suppose matrix \(A\) looks like this:

\[ A = \begin{bmatrix} a & b \ c & d \ e & f \ \end{bmatrix} \ \ \] then its transpose \(A^{\text{T}}\) would be:

\[ A^{\text{T}} = \begin{bmatrix} a & c & e \ b & d & f \ \ \end{bmatrix} \ \ \].

In notation, if a matrix \(A\) has elements \(A_{ij}\), where \(i\) represents the row and \(j\) represents the column, then its transpose \(A^{\text{T}}\) will have elements such that \((A^{\text{T}})_{ij} = A_{ji}\). This implies that the element in the \(i,j\)-th position of the original matrix becomes the element in the \(j,i\)-th position in the transpose matrix. This concept is especially useful when working with properties like symmetric matrices.

Matrix Multiplication

Matrix multiplication is another fundamental concept in linear algebra. When multiplying two matrices, the number of columns in the first matrix must equal the number of rows in the second matrix. For matrices \(A\) (of size \(m \times n\)) and \(B\) (of size \(n \times p\)), the resulting matrix \(C = A \times B\) will have dimensions \(m \times p\).

The element in the \(i, j\)-th position of the product matrix \(C\) is obtained by taking the dot product of the \(i\)-th row of \(A\) with the \(j\)-th column of \(B\). Mathematically, this can be written as:

\[ C_{ij} = \text{sum} (A_{ik} \times B_{kj} \text{, for all k}) \ \. \ \ \]

One crucial property of matrix multiplication used in this exercise is the transpose property:

\[ (AB)^{\text{T}} = B^{\text{T}}A^{\text{T}} \ \ \ \ , \ \ \]

This means that the transpose of a product of two matrices is equal to the product of the transposes of those matrices in reverse order. This property plays a key role in proving that \(AA^{\text{T}}\) is a symmetric matrix.

Linear Algebra

Linear algebra is a branch of mathematics focused on vectors, vector spaces, linear transformations, and systems of linear equations. Matrices, which are grids of numbers, are a crucial part of linear algebra. They help in representing and solving linear equations efficiently.

Symmetric matrices are an important concept in this field. A symmetric matrix is one that is equal to its transpose. Mathematically, a matrix \(C\) is symmetric if \(C = C^{\text{T}}\).

Symmetric matrices have numerous useful properties:

• They always have real eigenvalues.
• Their eigenvectors are orthogonal.
• They often appear in real-world applications like physics and statistics.

This specific exercise involves demonstrating that the product of a matrix and its transpose, \(A A^{\text{T}}\), is symmetric. Using properties from transpose and matrix multiplication, it's shown that \((A A^{\text{T}})^{\text{T}} = A A^{\text{T}}\) which confirms the symmetry. This is a practical example of how linear algebra techniques prove more complex properties of matrices.