Problem 17 Show that \(A+A^{T}\) is symmetr... [FREE SOLUTION]

Chapter 2: Problem 17

Show that \(A+A^{T}\) is symmetric for any square matrix \(A\).

Short Answer

Expert verified

\(A + A^{T}\) is symmetric because \(A + A^{T} = A^{T} + A\), showing self-equivalence to its transpose.

Step by step solution

Understand the Matrix Transpose

The transpose of a matrix, denoted as \(A^{T}\), is obtained by flipping the matrix over its diagonal, meaning the rows of \(A\) become the columns of \(A^{T}\).

Define Symmetric Matrix

A matrix is symmetric if it is equal to its transpose. Mathematically, matrix \(B\) is symmetric if \(B = B^{T}\).

Express \(A + A^{T}\) and Its Transpose

Consider the sum \(B = A + A^{T}\). Now, compute the transpose of \(B\): \(B^{T} = (A + A^{T})^{T} = A^{T} + (A^{T})^{T}\).

Simplify the Transpose

Recall that the transpose of a transpose returns the original matrix, \((A^{T})^{T} = A\). Thus, we have \(B^{T} = A^{T} + A\).

Compare \(B\) and \(B^{T}\)

Notice that \(B = A + A^{T}\) and \(B^{T} = A^{T} + A\). Since addition of matrices is commutative, \(A + A^{T} = A^{T} + A\).

Conclude Symmetry

Since \(B = B^{T}\), \(A + A^{T}\) is symmetric by definition, as it is equal to its transpose.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Transpose of a Matrix

To understand the concept of a transpose, think of flipping a matrix over its diagonal. The diagonal of a matrix is the line that runs from the top left corner to the bottom right corner. When you transpose a matrix, every element at position \((i, j)\) swaps with the element at position \((j, i)\).
This means that the rows become columns and the columns become rows.
For example: if you transpose a 2x2 matrix \(A = \begin{bmatrix} a & b \ c & d \end{bmatrix}\),you get \(A^{T} = \begin{bmatrix} a & c \ b & d \end{bmatrix}\).
Transposition is an important operation, especially when dealing with symmetry and matrix calculations.

Symmetric Matrix

A symmetric matrix is one where it looks the same even after transposing.
In simple terms, a matrix \(B\) is symmetric if \(B = B^{T}\).
This means that for all elements in the matrix, the element in the \((i, j)\) position is equal to the element in the \((j, i)\) position.

Symmetric matrices are always square, meaning they have the same number of rows and columns.
Mathematical operations with symmetric matrices often have special properties that simplify many proofs, such as in eigenvalues and orthogonal transformations.

When a symmetric matrix appears, certain matrix operations become more straightforward.

Matrix Addition Properties

Matrix addition is quite straightforward: it involves adding corresponding elements in the matrices.
But there are specific properties that make this operation consistent and useful:

Commutative Property: For any two matrices \(A\) and \(B\), \(A + B = B + A\). This means you can switch the order of matrices being added, and the result will not change.
Associative Property: For any matrices \(A\), \(B\), and \(C\), \((A + B) + C = A + (B + C)\). Grouping does not matter.
Identity Property: Adding a zero matrix (a matrix with all zero entries) to any matrix \(A\) will yield \(A\). So, \(A + 0 = A\).

These properties help simplify the manipulation of matrices and become particularly handy while proving results, such as showing that \(A + A^{T}\) is symmetric.

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