Math Problem Statement
are these matrices, $M(a,b,c)=\left(\begin{smallmatrix}
a&0&b&c\
0&a&c&b\
b&c&a&0\
c&b&0&a
\end{smallmatrix}\right)$, where $a,b,c\in\mathbb{R}$, diagonalizable?
Solution
To determine if the given matrix is diagonalizable, we need to check whether it has a full set of linearly independent eigenvectors. This can often be done by analyzing the eigenvalues and their corresponding eigenspaces.
Step 1: Matrix Structure
The given matrix is:
a & 0 & b & c \\ 0 & a & c & b \\ b & c & a & 0 \\ c & b & 0 & a \end{pmatrix}$$ where $$a, b, c$$ are real numbers. ### Step 2: Characteristic Polynomial To check diagonalizability, we can start by finding the eigenvalues, which are the roots of the characteristic polynomial $$p(\lambda) = \det(M - \lambda I)$$. The characteristic polynomial of a matrix $$M$$ is given by: $$\det(M - \lambda I) = \det\left(\begin{pmatrix} a - \lambda & 0 & b & c \\ 0 & a - \lambda & c & b \\ b & c & a - \lambda & 0 \\ c & b & 0 & a - \lambda \end{pmatrix}\right)$$ This determinant can be computed explicitly, but it involves a detailed calculation. ### Step 3: Symmetry and Possible Eigenvalues One way to simplify the analysis is to recognize the symmetry in the matrix. Specifically: - The matrix is symmetric (i.e., $$M = M^T$$). - Symmetric matrices over $$\mathbb{R}$$ are always diagonalizable because they have an orthogonal set of eigenvectors. ### Step 4: Eigenvalue Computation (Sketch) Given the symmetry, the matrix should be diagonalizable. However, let's outline how the eigenvalues might look: 1. For special cases, such as $$b = c = 0$$, the matrix is already diagonal. 2. For other cases, finding the characteristic polynomial will yield a quartic equation. Solving this quartic equation gives the eigenvalues, and the nature of these roots (distinct, repeated) will indicate the number of linearly independent eigenvectors. ### Conclusion Since $$M(a,b,c)$$ is symmetric, it is diagonalizable for any real values of $$a$$, $$b$$, and $$c$$. The matrix will have a full set of orthogonal eigenvectors corresponding to its eigenvalues. **Do you want details on how to explicitly compute the eigenvalues or have any questions?** --- **Related Questions:** 1. How do you compute the characteristic polynomial of a 4x4 matrix? 2. What are the properties of symmetric matrices in terms of eigenvalues and diagonalizability? 3. Can a matrix be diagonalizable if it has repeated eigenvalues? 4. What is the role of orthogonality in the diagonalization of symmetric matrices? 5. How does the structure of a matrix influence its diagonalizability? **Tip:** When dealing with symmetric matrices, remember they are always diagonalizable, which simplifies many problems involving eigenvalues and eigenvectors.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Algebra
Symmetric Matrices
Eigenvalues
Eigenvectors
Diagonalization
Formulas
Characteristic polynomial
Theorems
Diagonalizability of symmetric matrices
Suitable Grade Level
Advanced undergraduate level