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Chapter 2: Problem 8

Let \(A\) be a square matrix, and suppose that \(A=B+C\) where \(B\) is symmetric and \(C\) is antisymmetric. Show that $$ B=\frac{1}{2}\left(A+A^{\mathrm{T}}\right), \quad C=\frac{1}{2}\left(A-A^{\mathrm{T}}\right) $$

Short Answer

Expert verified
\(B = \frac{1}{2}(A + A^T), C = \frac{1}{2}(A - A^T)\) are the expressions for symmetric and antisymmetric parts of \(A\).

Step by step solution

01

Decompose the matrix expression

Given a square matrix \( A \) expressed as the sum of two matrices, \( A = B + C \), where \( B \) is symmetric and \( C \) is antisymmetric, we need to verify the proposed expressions for \( B \) and \( C \).
02

Understanding symmetric and antisymmetric properties

A symmetric matrix \( B \) satisfies \( B = B^T \), whereas an antisymmetric matrix \( C \) satisfies \( C = -C^T \). These properties will guide our calculations.
03

Find expression for \( B \)

To find \( B \), add \( A \) and its transpose: \( A + A^T = (B + C) + (B + C)^T = B + C + B^T + C^T \). Based on symmetry, \( C + C^T = 0 \), simplifying to \( A + A^T = 2B \). Therefore, solve for \( B \) as \( B = \frac{1}{2}(A + A^T) \).
04

Find expression for \( C \)

To solve for \( C \), subtract \( A^T \) from \( A \): \( A - A^T = (B + C) - (B + C)^T = B + C - B^T - C^T \). Using antisymmetry, \( B - B^T = 0 \), simplifying to \( A - A^T = 2C \). Hence, solve for \( C \) as \( C = \frac{1}{2}(A - A^T) \).
05

Verify both solutions

Check that both expressions satisfy the original matrix equation. Substitute back into the original: \( A = B + C \) where \( B = \frac{1}{2}(A + A^T) \) and \( C = \frac{1}{2}(A - A^T) \). Simplify to find \( A = \frac{1}{2}(A + A^T) + \frac{1}{2}(A - A^T) = A \), confirming the solution is correct.

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Key Concepts

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

Symmetric Matrix
A symmetric matrix is a square matrix that is equal to its transpose. Mathematically, if we denote a matrix as \( B \), then it is symmetric if \( B = B^T \). This property implies that the matrix is a mirror image of itself across the diagonal.
  • Simplifying this, each element above the diagonal is equal to its corresponding element below the diagonal.
  • It is crucial in various mathematical contexts as it often simplifies computations due to its unique symmetry property.
  • Applications of symmetric matrices include optimization problems and in statistics, such as in covariance matrices.
Whether it's an application in physics with concepts like inertia tensor or in machine learning with kernel functions, symmetric matrices are deeply integrated into scientific calculations.
Antisymmetric Matrix
An antisymmetric matrix, also known as a skew-symmetric matrix, is a square matrix that satisfies the condition that it is equal to the negative of its transpose. For a matrix \( C \), this means \( C = -C^T \).
  • This definition implies that all diagonal elements of an antisymmetric matrix are zero because for any diagonal element we have \( c_{ii} = -c_{ii} \), which results in \( c_{ii} = 0 \).
  • Antisymmetric matrices have properties that make them useful in areas such as vector algebra and are used to represent cross products in three-dimensional space.
  • In physics, they appear in the representation of angular momentum and other rotational dynamics.
Understanding these properties helps simplify many mathematical problems, especially those involving transformations and rotations.
Matrix Decomposition
Matrix decomposition refers to the process of breaking down a matrix into simpler, more manageable parts, typically to simplify complex matrix operations. The idea is to express a matrix as a product or sum of simpler matrices.
  • This process is essential in numerical analysis and linear algebra, providing a foundation for algorithms solving system of equations, inverse matrix calculations, and more.
  • Decomposition can take various forms, including LU decomposition, QR decomposition, and spectral decomposition, each serving different kinds of problems.
  • In the context of this exercise, decomposing matrix \( A \) into the sum of symmetric \( B \) and antisymmetric \( C \) provides insight into its structure and properties.
This particular form of decomposition helps in understanding the structural aspects of matrices and solving linear problems efficiently.
Matrix Transpose
The transpose of a matrix is an operation that flips the elements of a matrix over its main diagonal. If you have a matrix \( A \), its transpose, denoted \( A^T \), is achieved by swapping rows with columns.
  • If \( A \) is of size \( m \times n \), then \( A^T \) will be \( n \times m \).
  • This transformation is vital in a wide range of concepts including solving system of equations and transformation of matrices in mathematical computations.
  • Transpose operations are often used in matrix equations and are a key concept when dealing with symmetric and antisymmetric matrices, as they directly affect the symmetry properties of the matrix.
Understanding how to transpose a matrix and its implications on matrix properties is fundamental to mastering concepts in linear algebra.

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Most popular questions from this chapter

For each of the following, decide whether it is defined and, if it is, compute the result: $$ \left[\begin{array}{l} 1 \\ 2 \end{array}\right]+\left[\begin{array}{ll} 3 & 2 \end{array}\right] $$

Let \(A\) be an \(n \times n\) matrix for which there is exactly one matrix \(B\) such that \(A B=I\). By considering \(B A+B-I\) show that \(A\) is invertible.

For each of the following, decide whether it is defined and, if it is, compute the result: $$ i\left[\begin{array}{ll} i & 1-i \end{array}\right]-(1+i)\left[\begin{array}{ll} 1 & -i \end{array}\right] \quad \mathbf{1 2} \quad 2\left[\begin{array}{r} 0 \\ -1 \end{array}\right]-3\left[\begin{array}{l} -1 \\ -2 \end{array}\right] $$

Let \(A, B, C, D\) be the following matrices: $$ A=\left[\begin{array}{rr} 1 & 2 \\ 4 & -3 \end{array}\right], \quad B=\left[\begin{array}{ll} 0 & 2 \\ 3 & 4 \\ 1 & 2 \end{array}\right], \quad C=\left[\begin{array}{ll} 1 & 4 \end{array}\right], \quad D=\left[\begin{array}{l} 4 \\ 1 \end{array}\right] \text {. } $$ Which of the products \(A^{2}, A B, A C, A D, B A, B^{2}\), etc. exist? Evaluate the products that do exist.

Write down the sizes of the following matrices: (a) \(\left[\begin{array}{cc}i & -i \\ 1 & 2 \\ 1+i & 4 \\ \frac{1}{2} & -1\end{array}\right]\) (b) \(\left[\begin{array}{llllll}1 & -1 & 1 & -1 & 1 & -1\end{array}\right]\) (c) \(\left[\begin{array}{rrr}1 & 3 & 5 \\ 2 & -1 & -4 \\ 6 & 1 & 3\end{array}\right]\) (d) \(\left[\begin{array}{rr}1 & -1 \\ -1 & 1\end{array}\right]\).
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