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Suppose $A$ is a symmetric $n \times n$ matrix and $B$ is any $n \times m$ matrix. Show that $B^{T} A B, B^{T} B,$ and $B B^{T}$ are symmetric matrices.

since $A$ is symmetric, $\left(B^{T} A B\right)^{T}=B^{T} A^{T} B^{T T}=B^{T} A B,$ and $B^{T} A B$ issymmetric. Applying this result with $A=I$ gives $B^{T} B$ is symmetric. Finally,$\left(B B^{T}\right)^{T}=B^{T T} B^{T}=B B^{T},$ so $B B^{T}$ is symmetric.

Algebra

Chapter 7

Symmetric Matrices and Quadratic Forms

Section 1

Diagonalization of Symmetric Matrices

Introduction to Matrices

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