91影视

• In linear algebra, a symmetric matrix is a square matrix that does not change when its transpose is calculated.
• A symmetric matrix is defined as one whose transpose is identical to the matrix itself.
• A square matrix of size $\mathit{n}\mathbf{脳}\mathit{n}$is symmetric if${\mathit{B}}^{\mathbf{T}}\mathbf{=}\mathit{B}$ .

We can consider $l_3$to be a degenerated reflection. Now, here we have,

$$\begin{gathered} \stackrel{鈫�}{v}=l_3\stackrel{鈫�}{v}鈫�\stackrel{鈫�}{v}=A^2\stackrel{鈫�}{v} \\ \stackrel{鈫�}{v}={位}^2\stackrel{鈫�}{v} \\ \stackrel{鈫�}{v}={位}^2\stackrel{鈫�}{v} \\ {位}^2=1 \\ 位=卤1 \end{gathered}$$

where $位=1$and $位=-1$. Now, since the matrix A is symmetric, the eigenspaces$E_1$ and $E_{-1}$ corresponding to the eigenvalue $位=1$and $位=-1$respectively, are orthogonal complements.

The eigenspace associated with the eigenvalue $位=1$is exactly the space about which we are doing the reflection.

Now, vectors that are orthogonal to$E_1$ are mapped on their opposite.

Therefore, the linear transformation $T\left(\stackrel{鈫�}{x}\right)=A\stackrel{鈫�}{x}$is a reflection about the subspace of ${\mathrm{鈩�}}^3{\mathrm{鈩�}}^3$.