\[ \beta = \{v_1, \cdots, v_n\} \]

\[ \beta = \{v_1, \cdots, v_n\} \]

• \(v_i ^ T v_j = 0\), \(i \neq j\)
• \(v_i^T v_i = 1\)

\[ Q = \begin{bmatrix} \vert & & \vert\\ v_1 & \cdots & v_n\\ \vert & & \vert \end{bmatrix} \]

\[ \beta = \{v_1, \cdots, v_n\} \]

• \(v_i ^ T v_j = 0\), \(i \neq j\)
• \(v_i^T v_i = 1\)

\[ Q = \begin{bmatrix} \vert & & \vert\\ v_1 & \cdots & v_n\\ \vert & & \vert \end{bmatrix} \]
\[ Q^T Q = \begin{bmatrix} - & v_1 & -\\ & \vdots & \\ - & v_n & - \end{bmatrix} \begin{bmatrix} \vert & & \vert\\ v_1 & \cdots & v_n\\ \vert & & \vert \end{bmatrix} \]

\[ \beta = \{v_1, \cdots, v_n\} \]

• \(v_i ^ T v_j = 0\), \(i \neq j\)
• \(v_i^T v_i = 1\)

\[ Q = \begin{bmatrix} \vert & & \vert\\ v_1 & \cdots & v_n\\ \vert & & \vert \end{bmatrix} \]
\[ Q^T Q = I \]

\[ \beta = \{v_1, \cdots, v_n\} \]

• \(v_i ^ T v_j = 0\), \(i \neq j\)
• \(v_i^T v_i = 1\)

\[ Q = \begin{bmatrix} \vert & & \vert\\ v_1 & \cdots & v_n\\ \vert & & \vert \end{bmatrix} \]
\[ Q^T Q = I \]
\[ Q^{-1} = Q^T \]

\[ A = QDQ^{-1} \]

\[ A = QDQ^T \]

\[ A = QDQ^T \]

\[ A^T = \]

\[ A = QDQ^T \]

\[ A^T = A \]

If a matrix is orthogonally diagonalizable, it is symmetric.

If a matrix is orthogonally diagonalizable, it is symmetric.

If a matrix is symmetric, is it orthogonally diagonalizable?

If a matrix is orthogonally diagonalizable, it is symmetric.

If a matrix is symmetric, is it orthogonally diagonalizable?

Week-5