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Chapter 7: Problem 20

Prove: If \(\left\\{\mathbf{u}_{1}, \mathbf{u}_{2}, \ldots, \mathbf{u}_{n}\right\\}\) is an orthonormal basis for \(R^{n},\) and if \(A\) can be expressed as $$A=c \mathbf{u}_{1} \mathbf{u}_{1}^{T}+c_{2} \mathbf{u}_{2} \mathbf{u}_{2}^{T}+\ldots+c_{n} \mathbf{u}_{n} \mathbf{u}_{n}^{T}$$ then \(A\) is symmetric and has eigenvalues \(c_{1}, c_{2}, \ldots, c_{n}.\)

Short Answer

Expert verified
Matrix \( A \) is symmetric with eigenvalues \( c_1, c_2, \ldots, c_n \).

Step by step solution

01

Understanding the Orthogonal Matrix

First, we recognize that the vectors \( \{ \mathbf{u}_1, \mathbf{u}_2, \ldots, \mathbf{u}_n \} \) form an orthonormal basis. This means that these vectors are not only orthogonal to each other, but also each have a unit length. The orthogonality implies \( \mathbf{u}_i^T \mathbf{u}_j = 0 \) for \( i eq j \), and \( \mathbf{u}_i^T \mathbf{u}_i = 1 \).
02

Verifying Matrix A is Symmetric

A matrix \( A \) is symmetric if \( A = A^T \). Now consider the expression for \( A \): \( A = c_1 \mathbf{u}_1 \mathbf{u}_1^T + c_2 \mathbf{u}_2 \mathbf{u}_2^T + \ldots + c_n \mathbf{u}_n \mathbf{u}_n^T \). Each individual term \( c_i \mathbf{u}_i \mathbf{u}_i^T \) is symmetric because the outer product with itself forms a symmetric matrix, i.e., \( (c_i \mathbf{u}_i \mathbf{u}_i^T)^T = c_i (\mathbf{u}_i^T)^T \mathbf{u}_i^T = c_i \mathbf{u}_i \mathbf{u}_i^T \). Therefore, the sum of symmetric matrices is symmetric.
03

Eigenvalues Determination

To determine the eigenvalues, we use the fact that \( A \mathbf{u}_i = (c_1 \mathbf{u}_1 \mathbf{u}_1^T + c_2 \mathbf{u}_2 \mathbf{u}_2^T + \ldots + c_n \mathbf{u}_n \mathbf{u}_n^T) \mathbf{u}_i \). Due to the orthonormality, \( \mathbf{u}_j^T \mathbf{u}_i = 0 \) for \( j eq i \), this simplifies to \( A \mathbf{u}_i = c_i \mathbf{u}_i \). Thus, each vector \( \mathbf{u}_i \) is an eigenvector corresponding to eigenvalue \( c_i \).
04

Conclusion

Each term \( c_i \mathbf{u}_i \mathbf{u}_i^T \) confirms that \( \mathbf{u}_i \) is associated with the eigenvalue \( c_i \), forming the spectrum of \( A \). The matrix \( A \) is symmetric and its eigenvalues are precisely the coefficients \( c_1, c_2, \ldots, c_n \).

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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. This means if you flip the matrix over its diagonal, it looks the same. In other words, a matrix \( A \) is symmetric if \( A = A^T \). The symmetry can be seen in simple terms: the element in the i-th row and j-th column is the same as the element in the j-th row and i-th column. This property is crucial in many areas of linear algebra and has important implications:
  • Symmetric matrices always have real eigenvalues.
  • The eigenvectors of symmetric matrices can be chosen to be orthogonal, which means they form an orthonormal basis, simplifying many computations.
  • In a symmetric matrix, if one element changes value, the corresponding element must change to maintain symmetry.
Understanding how symmetric matrices work helps in solving problems involving eigenvalues and eigenvectors, where symmetry leads to simplified solutions.
Eigenvalues
Eigenvalues are a fundamental concept in linear algebra, often arising in equations involving matrices. Given a square matrix \( A \), an eigenvalue \( \lambda \) is a number such that there is a non-zero vector \( \mathbf{v} \) (the eigenvector) satisfying the equation \( A\mathbf{v} = \lambda\mathbf{v} \). In simpler terms, when the matrix \( A \) acts on the vector \( \mathbf{v} \), the result is just a scaled version of \( \mathbf{v} \).
  • Each symmetric matrix has eigenvectors that are orthogonal, and these eigenvectors can be used to construct an orthonormal basis.
  • The eigenvalues reveal insights into the matrix's properties, such as stability or geometric transformations like rotations or scalings.
  • Finding the eigenvalues of a matrix involves solving the characteristic equation \( \det(A - \lambda I) = 0 \), where \( I \) is the identity matrix.
In practical applications, these concepts are crucial in fields like physics, engineering, and computer science, where they are used to analyze systems and transformations.
Orthogonal Matrix
An orthogonal matrix is a square matrix whose rows and columns are orthonormal vectors. This means each vector is unit length and orthogonal to every other vector. Mathematically, a matrix \( Q \) is orthogonal if \( Q^T Q = Q Q^T = I \), where \( I \) is the identity matrix. Here are some important properties:
  • Orthogonal matrices preserve lengths and angles, making them valuable in preserving vector norms during transformations.
  • The determinant of an orthogonal matrix is always \( \pm 1 \).
  • Orthogonal matrices are symmetric if the matrix is real, which aligns with our understanding of eigenvectors forming an orthonormal set.
Such matrices are often employed in numerical analysis and computer graphics for tasks that require stable and accurate rotations or reflections.
Linear Algebra
Linear algebra is the branch of mathematics that studies vectors, vector spaces, linear mappings, and systems of linear equations. Many concepts in linear algebra, such as matrices and determinants, are widely applicable in areas ranging from engineering to data science.
  • Matrices are central to linear algebra and serve as compact representations of linear transformations.
  • Understanding the properties of matrices, including their eigenvalues and eigenvectors, allows for the solving of practical problems involving systems of equations.
  • Linear algebra also includes study of vector spaces and orthonormal bases, essential for modeling multidimensional data and transformations.
By providing the language and framework to describe linear combinations and transformations, linear algebra underpins many facets of both theoretical and applied sciences.

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Most popular questions from this chapter

Verify that the eigenvalues of the Hermitian matrix \(A\) are real and that eigenvectors from different eigenspaces are orthogonal (see Theorem 7.5.3). $$A=\left[\begin{array}{cc}3 & 2-3 i \\\2+3 i & -1\end{array}\right]$$

Classify the quadratic form as positive definite, negative definite, indefinite, positive semidefinite, or negative semidefinite. $$-\left(x_{1}-x_{2}\right)^{2}$$

Find a formula for the quadratic form that does not use matrices. $$\left[\begin{array}{lll} x_{1} & x_{2} & x_{3} \end{array}\right]\left[\begin{array}{ccc} -2 & \frac{7}{2} & 1 \\ \frac{7}{2} & 0 & 6 \\ 1 & 6 & 3 \end{array}\right]\left[\begin{array}{l} x_{1} \\ x_{2} \\ x_{3} \end{array}\right]$$

Under what conditions is the following matrix normal? $$A=\left[\begin{array}{lll}a & 0 & 0 \\ 0 & 0 & c \\\0 & b & 0\end{array}\right]$$

Prove: If \(b \neq 0,\) then the cross product term can be eliminated from the quadratic form \(a x^{2}+2 b x y+c y^{2}\) by rotating the coordinate axes through an angle \(\theta\) that satisfies the equation $$\cot 2 \theta=\frac{a-c}{2 b}.$$
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