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Chapter 3: Problem 35

(a) Prove that if \(A\) and \(B\) are symmetric \(n \times n\) matrices, then so is \(A+B\). (b) Prove that if \(A\) is a symmetric \(n \times n\) matrix, then so is \(k A\) for any scalar \(k\).

Short Answer

Expert verified
(a) \(A+B\) is symmetric because \((A+B)^T = A^T + B^T = A + B\). (b) \(kA\) is symmetric because \((kA)^T = kA^T = kA\).

Step by step solution

01

Understanding Symmetric Matrices

A matrix \(M\) is symmetric if \(M = M^T\), where \(M^T\) is the transpose of \(M\). This means that the elements across the diagonal are equal: \(m_{ij} = m_{ji}\) for all \(i\) and \(j\).
02

Proving A + B is Symmetric

Given that both matrices \(A\) and \(B\) are symmetric, we have \(A = A^T\) and \(B = B^T\). To prove that \(A + B\) is symmetric, we need to show that \((A + B)^T = A + B\).Taking the transpose of \(A + B\), we use the property of the transpose of a sum: \((A + B)^T = A^T + B^T\). Since \(A\) and \(B\) are symmetric matrices, \(A^T = A\) and \(B^T = B\). Thus, \((A + B)^T = A + B\), proving \(A + B\) is symmetric.
03

Proving kA is Symmetric

Given that matrix \(A\) is symmetric, we know \(A = A^T\). To prove that \(kA\) is symmetric, we need to show that \((kA)^T = kA\).The transpose of a scalar multiple of a matrix \(kA\) is \((kA)^T = kA^T\). Because \(A\) is symmetric, \(A^T = A\). Hence, \((kA)^T = kA\), confirming that \(kA\) is symmetric for any scalar \(k\).

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Key Concepts

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

Matrix Transpose
The transpose of a matrix is a fundamental operation in linear algebra. When you take the transpose of a matrix, you are essentially "flipping" it over its diagonal.
  • In a matrix denoted by \( M \), the transpose is written as \( M^T \).
  • Element \( m_{ij} \) in the original matrix becomes element \( m_{ji} \) in the transposed matrix.
  • This means rows become columns, and vice versa.
To visualize this, imagine a simple 2x2 matrix:\[M = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]The transpose of \( M \) would be: \[M^T = \begin{pmatrix} a & c \ b & d \end{pmatrix}\]In our context of symmetric matrices, an important property emerges; a matrix \( M \) is symmetric if \( M = M^T \). This property is critical in proving other mathematical truths about matrices, such as demonstrating the symmetry in matrix operations like addition or scalar multiplication.
Matrix Addition
Matrix addition is a basic operation that involves adding corresponding elements of two matrices together. It is as simple as it sounds but requires both matrices to be of the same size.
  • If you have matrices \( A \) and \( B \), both of dimension \( n \times n \), their sum \( A + B \) is constructed by adding their respective elements.
  • The element at position \( (i, j) \) in \( A + B \) is \( a_{ij} + b_{ij} \).
Suppose \( A \) and \( B \) are both 2x2 symmetric matrices:\[A = \begin{pmatrix} a & b \ b & c \end{pmatrix}, \quad B = \begin{pmatrix} e & f \ f & g \end{pmatrix}\]Their sum would be:\[A + B = \begin{pmatrix} a+e & b+f \ b+f & c+g \end{pmatrix}\]The resulting matrix is also symmetric because for all \( i, j \), \( a_{ij} = a_{ji} \) holds. This demonstrates why the property \( (A+B) = (A+B)^T \) is essential when dealing with symmetric matrices.
Scalar Multiplication
Scalar multiplication in linear algebra involves multiplying each element of a matrix by a constant, known as a scalar.
  • Given a scalar \( k \) and a matrix \( A \), the product \( kA \) involves multiplying each element in \( A \) by \( k \).
  • This operation is applied uniformly across all elements of the matrix.
Consider a 2x2 matrix \( A \):\[A = \begin{pmatrix} a & b \ c & d \end{pmatrix}\]If we multiply \( A \) by a scalar \( k \), the result would be:\[kA = \begin{pmatrix} ka & kb \ kc & kd \end{pmatrix}\]This operation maintains the symmetry of matrix \( A \) because switching rows and columns does not change the proportional relationships between them. Hence, if \( A \) is initially symmetric, \( kA \) remains symmetric. This forms the core idea behind why scalar multiplication preserves symmetry in matrices.
Linear Algebra
Linear Algebra is a crucial field of mathematics focused on vector spaces and the transformations between them.
  • It deals with structures such as matrices and vectors, along with the operations that can be performed on them.
  • Concepts such as matrix transpose, matrix addition, and scalar multiplication are foundational to understanding linear algebra.
These operations are not merely computational; they hold significant theoretical significance in subjects like computer science, physics, and engineering. The study of symmetric matrices and their properties is a significant area within linear algebra.
  • Understanding symmetry in matrices helps in simplifying many problems, including solutions to linear systems and eigenvalue calculations.
  • Symmetric matrices have eigenvalues that are real and eigenvectors that can be chosen to be orthogonal.
This makes them extremely useful in applications such as optimization problems and computer graphics. In essence, linear algebra provides the framework and language for much of modern science and mathematics.

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Most popular questions from this chapter

Verify Theorem 3.32 by finding the matrix of \(S\) o \(T\) (a) by direct substitution and (b) by matrix multiplication of \([\mathrm{S}][\mathrm{T}]\). $$T\left[\begin{array}{l} x_{1} \\ x_{2} \end{array}\right]=\left[\begin{array}{c} x_{1}+2 x_{2} \\ -3 x_{1}+x_{2} \end{array}\right], S\left[\begin{array}{l} y_{1} \\ y_{2} \end{array}\right]=\left[\begin{array}{c} y_{1}+3 y_{2} \\ y_{1}-y_{2} \end{array}\right]$$

A graph is called bipartite if its vertices can be subdivided into two sets U and V such that every edge has one endpoint in \(U\) and the other endpoint in \(V\). For example, the graph in Exercise 46 is bipartite with \(U=\left\\{v_{1}, v_{2}, v_{3}\right\\}\) and \(V=\left\\{v_{4}, v_{5}\right\\}\). Determine whether \(a\) graph with the given adjacency matrix is bipartite. $$\left[\begin{array}{cccccc} 0 & 0 & 1 & 0 & 1 & 1 \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 & 1 & 1 \\ 1 & 1 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 1 & 0 & 0 \end{array}\right]$$

Draw a graph that has the given adjacency matrix. $$\left[\begin{array}{lllll} 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 & 1 \\ 1 & 1 & 1 & 0 & 0 \\ 1 & 1 & 1 & 0 & 0 \end{array}\right]$$

Find the standard matrix of the composite transformation from \(\mathbb{R}^{2}\) to \(\mathbb{R}^{2}\). Clockwise rotation through \(45^{\circ},\) followed by projection onto the \(y\) -axis, followed by clockwise rotation through \(45^{\circ}\).

Use the Gauss-Jordan method to find the inverse of the given matrix (if it exists). $$\left[\begin{array}{rrrr}0 & -1 & 1 & 0 \\\2 & 1 & 0 & 2 \\\1 & -1 & 3 & 0 \\\0 & 1 & 1 & -1\end{array}\right]$$
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