For each of the matrices A in Exercises 7 through 11, find...

For each of the matrices $A$ in Exercises 7 through $11,$ find an orthogonal matrix S and a diagonal matrix $D$ such that $S^{-1} A S=D .$ Do not use technology.
$$A=\left[\begin{array}{lll}0 & 0 & 3 \\0 & 2 & 0 \\3 & 0 & 0\end{array}\right]$$

Key Concepts

Matrix Diagonalization

Diagonalization is the process of finding a diagonal matrix that is similar to the original matrix, meaning the original matrix can be expressed as S?¹AS = D where S contains the eigenvectors and D contains the eigenvalues. This process simplifies many matrix computations, such as finding powers of matrices, and is a powerful tool in many areas of applied mathematics and physics.

Orthogonal Matrices

An orthogonal matrix is one whose columns form an orthonormal set, meaning each column vector has unit length and is perpendicular to the others. Orthogonal matrices preserve angles and lengths, and their inverses are simply their transposes. They are used in diagonalization to ensure numerical stability and to simplify computations, especially when converting a symmetric matrix into a diagonal form.

Eigenvalues and Eigenvectors

These are central concepts in linear algebra for understanding the action of a matrix. Eigenvalues are scalars that scale eigenvectors, which are nonzero vectors that only change by a scalar factor when a matrix is applied to them. In the context of diagonalization, the eigenvalues appear on the diagonal of the resulting matrix, and the associated eigenvectors form the columns of the change-of-basis matrix.

Spectral Theorem

The spectral theorem states that every real symmetric matrix can be diagonalized by an orthogonal matrix. This means that there exists an orthogonal matrix whose columns are eigenvectors of the symmetric matrix, and the diagonal matrix will have the corresponding eigenvalues on its diagonal. This theorem is fundamental because it guarantees both the existence of a full set of real eigenvalues and an orthonormal basis of eigenvectors for symmetric matrices.

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