In Exercises 25 and 26, mark each statement True or False....

In Exercises 25 and $26,$ mark each statement True or False. Justify each answer. a. There are symmetric matrices that are not orthogonally diagonalizable. b. If $B=P D P^{T}$ , where $P^{T}=P^{-1}$ and $D$ is a diagonal matrix, then $B$ is a symmetric matrix. c. An orthogonal matrix is orthogonally diagonalizable. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

In Exercises 25 and $26,$ mark each statement True or False. Justify each answer.
a. There are symmetric matrices that are not orthogonally diagonalizable.
b. If $B=P D P^{T}$ , where $P^{T}=P^{-1}$ and $D$ is a diagonal
matrix, then $B$ is a symmetric matrix.
c. An orthogonal matrix is orthogonally diagonalizable. The dimension of an eigenspace of a symmetric matrix is sometimes less than the multiplicity of the corresponding eigenvalue.

Key Concepts

Symmetric Matrices

A symmetric matrix is one that is equal to its own transpose. This property guarantees that all of its eigenvalues are real and that the matrix has a full set of orthonormal eigenvectors. In linear algebra, symmetric matrices exhibit many favorable properties, including being amenable to diagonalization through an orthogonal change of basis.

Orthogonal Diagonalization

Orthogonal diagonalization is a special form of diagonalization where the change of basis matrix is orthogonal, meaning its transpose is also its inverse. This process simplifies computations and preserves the inner product structure of the space. It is particularly important for symmetric matrices, as the Spectral Theorem ensures that every symmetric matrix can be orthogonally diagonalized.

Spectral Theorem

The Spectral Theorem is a fundamental result in linear algebra which states that every real symmetric matrix can be diagonalized by an orthogonal matrix. This theorem underpins many techniques in mathematics and engineering, ensuring that symmetric matrices have real eigenvalues and a complete set of orthonormal eigenvectors, thereby making them especially tractable for analysis.

Orthogonal Matrices

An orthogonal matrix is a square matrix with the property that its transpose is equal to its inverse. This implies that the columns (and rows) of the matrix form an orthonormal basis, preserving lengths and angles under transformation. Orthogonal matrices are central to many areas of linear algebra, including the process of orthogonal diagonalization.

Eigenvalue Multiplicity and Eigenspaces

The concept of eigenvalue multiplicity involves two related ideas: the algebraic multiplicity, which is the number of times an eigenvalue appears as a root of the characteristic polynomial, and the geometric multiplicity, which is the dimension of the corresponding eigenspace. For symmetric matrices, the Spectral Theorem guarantees that the geometric multiplicity equals the algebraic multiplicity, meaning that the eigenspace of each eigenvalue is as large as the eigenvalue's multiplicity.

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