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Orthogonal diagonalization of a symmetric matrix is a diagonalization by an orthogonal shift of coordinates in linear algebra.

A matrix is said to be symmetric if an orthogonal matrix can diagonalize it.

Equal numbers of rows and columns make up a square matrix. If the transpose of a square matrix with real numbers or elements equals the inverse matrix, the matrix is said to be orthogonal. Or, we may argue that a square matrix is an orthogonal matrix if the product of the square matrix and its transpose results in an identity matrix.

Square matrix A is orthogonal:

$$\begin{gathered} A^T=A^{-1}=I \\ {A}^T=TransposeofA \\ {A}^{-1}=InverseofA \end{gathered}$$

I = Identity matrix of same order as A

The matrix of rotation in $R^2$ is an orthogonal matrix but does not have real eigenvalues and hence is not diagonalizable.

Thus, the given statement is FALSE.