Problem 18 - Advanced Engineering Mathematics

Question

In Problems 11-18, proceed as in Example 3 to construct an orthogonal matrix from the eigenvectors of the given symmetric matrix. (The answers are not unique.) $$ \left(\begin{array}{rrr} 2 & 8 & -2 \\ 8 & -4 & 10 \\ -2 & 10 & -7 \end{array}\right) $$

Step-by-Step Solution

Answer

Find eigenvalues and eigenvectors, normalize eigenvectors, and form the orthogonal matrix using these vectors.

Step 1: Find the Eigenvalues

For a symmetric matrix \(A\), the eigenvalues can be found by solving the characteristic polynomial \( \det(A - \lambda I) = 0 \). Here, \(A\) is \(\begin{pmatrix} 2 & 8 & -2 \ 8 & -4 & 10 \ -2 & 10 & -7 \end{pmatrix}\) and \(I\) is the identity matrix. Calculate the determinant of \(A - \lambda I\): \[ \det \begin{pmatrix} 2-\lambda & 8 & -2 \ 8 & -4-\lambda & 10 \ -2 & 10 & -7-\lambda \end{pmatrix} = 0 \] Expand the determinant to find the characteristic polynomial and solve for \(\lambda\).

Step 2: Solve the Characteristic Polynomial

After expanding the determinant, you get a cubic polynomial equation in terms of \(\lambda\). Find the roots of this polynomial which are the eigenvalues of the matrix. Typically in these exercises, they are integers or simple fractions.

Step 3: Find Corresponding Eigenvectors

For each eigenvalue \(\lambda_i\) found in the previous step, find the eigenvectors by solving the equation \((A - \lambda_i I)\mathbf{x} = \mathbf{0}\) for \(\mathbf{x}\). This involves row-reducing the matrix \(A - \lambda_i I\) and finding the null space.

Step 4: Normalize the Eigenvectors

Ensure each eigenvector has unit length by dividing each vector by its norm. The norm of a vector \(\mathbf{v}\) is \(\|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2 + v_3^2}\). Use this to scale each vector to unit length.

Step 5: Construct the Orthogonal Matrix

Combine the normalized eigenvectors as columns to form the orthogonal matrix. Since the eigenvectors are orthogonal and normalized, this matrix will be an orthogonal matrix, meaning its transpose equals its inverse, \( Q^T = Q^{-1} \).

Step 6: Verify Orthogonality

Confirm the matrix is orthogonal by checking that \(Q^T Q = I\). This ensures the correctness of the constructed orthogonal matrix from the normalized eigenvectors.

Key Concepts

Eigenvectors

When dealing with matrices, you might come across the concept of eigenvectors. An eigenvector is a nonzero vector that changes at most by a scalar factor when a linear transformation is applied to it. In simpler terms, if we have a matrix \(A\) and a vector \(\mathbf{x}\), then \(\mathbf{x}\) is an eigenvector of \(A\) if it satisfies the equation \(A\mathbf{x} = \lambda \mathbf{x}\). Here, \(\lambda\) is what we call an eigenvalue, which we'll talk about in more detail later.

• Eigenvectors give us valuable insights into the properties of the transformation represented by the matrix.
• Finding eigenvectors helps in simplifying matrix computations, such as matrix diagonaliation.
• Eigenvectors are part of constructing orthogonal matrices, which have numerous applications.

In practice, we determine eigenvectors by first finding eigenvalues, and then using them to solve a system of linear equations derived from \((A - \lambda I)\mathbf{x} = \mathbf{0}\). This process involves some computation, but once you have your eigenvalues, finding the corresponding eigenvectors becomes straightforward.

Symmetric Matrix

A symmetric matrix is a special kind of square matrix. It is characterized by its symmetry about its main diagonal, meaning that it mirrors itself. Mathematically speaking, a matrix \(A\) is symmetric if \(A = A^T\) (where \(A^T\) is the transpose of \(A\)).

• One of the most interesting properties of symmetric matrices is that all of their eigenvalues are real numbers.
• When a matrix is symmetric, you can be sure there is a complete set of orthogonal eigenvectors.
• This leads to easier computations particularly in constructing orthogonal matrices, important in various branches of mathematics and physics.

Symmetric matrices are prevalent due to their stability and their occurrence in many real-world applications, like structural analysis, physics, engineering, and computer graphics. Their properties simplify many computational problems, especially those involving eigenvalues and eigenvectors.

Eigenvalues

Eigenvalues are intrinsic to understanding so many phenomena in mathematics. An eigenvalue associated with a square matrix \(A\) is a scalar, \(\lambda\), such that when it multiplies the identity matrix \(I\), the matrix \(A - \lambda I\) becomes singular. This means its determinant is zero:
\[ \det(A - \lambda I) = 0 \]

• This equation is known as the characteristic equation, and solving it involves finding the roots of the resulting polynomial.
• For symmetric matrices, all eigenvalues are guaranteed to be real numbers, making them easier to handle computationally.
• The roots provide the eigenvalues needed for finding eigenvectors and constructing orthogonal matrices.

Understanding eigenvalues is crucial because they provide information on matrix properties like stability and oscillation in systems, which is fundamental in dynamic systems and signal processing.

Characteristic Polynomial

Characteristic polynomials are central to finding a matrix's eigenvalues. When you subtract \(\lambda I\) (where \(I\) is the identity matrix) from your original matrix \(A\), you obtain a new matrix \((A - \lambda I)\), whose determinant expands into a polynomial in terms of \(\lambda\). This polynomial, \(p(\lambda) = \det(A - \lambda I)\), is what we call the characteristic polynomial.

• The degree of the characteristic polynomial equals the size of the matrix you started with.
• Solving \(p(\lambda) = 0\) gives you the eigenvalues of \(A\).
• For a 3x3 matrix, such as the one in the exercise, \(p(\lambda)\) will be a cubic polynomial.

Understanding and being able to compute the characteristic polynomial is crucial, as its roots are directly the eigenvalues of the matrix, leading you to finding eigenvectors and constructing important matrix forms like orthogonal matrices.