Is a symmetric matrix always orthogonally diagonalizable?

The amazing thing is that the converse is also true: Every real symmetric matrix is orthogonally diagonalizable. The proof of this is a bit tricky. However, for the case when all the eigenvalues are distinct, there is a rather straightforward proof which we now give.

Can a matrix be symmetric and orthogonal?

Orthogonal matrices are square matrices with columns and rows (as vectors) orthogonal to each other (i.e., dot products zero). An orthogonal matrix is symmetric if and only if it’s equal to its inverse.

Why are symmetric matrices orthogonally diagonalizable?

The Spectral Theorem: A square matrix is symmetric if and only if it has an orthonormal eigenbasis. Equivalently, a square matrix is symmetric if and only if there exists an orthogonal matrix S such that ST AS is diagonal. That is, a matrix is orthogonally diagonalizable if and only if it is symmetric.

Can an orthogonal matrix be orthogonally diagonalizable?

(b) An orthogonal matrix is always orthogonally diagonalizeable.

Which symmetric matrices are also orthogonal?

Symmetric matrices with n distinct eigenvalues are orthogonally diagonalizable. since a and b are distinct, we can conclude that v and w are orthogonal. a fact that is left for you as an exercise. There is special property that holds for orthogonal matrices that is worth noting.

Is the difference of symmetric matrices symmetric?

Properties of Symmetric Matrix Addition and difference of two symmetric matrices results in symmetric matrix. If A and B are two symmetric matrices and they follow the commutative property, i.e. AB =BA, then the product of A and B is symmetric. If matrix A is symmetric then An is also symmetric, where n is an integer.

Are orthogonal matrices orthogonally diagonalizable?

Do symmetric matrices have orthogonal eigenvectors?

Symmetric Matrices A has exactly n (not necessarily distinct) eigenvalues. There exists a set of n eigenvectors, one for each eigenvalue, that are mututally orthogonal.