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I'm just trying to clear a few things up about diagonalising a matrix. As I understand it, if you have a Euclidean space and a symmetric matrix, there is essentially two ways of diagonalising this matrix.

One is for the quadratic form generated by this matrix, finding an orthogonal basis which makes the matrix for this quadratic form a diagonal. This is the same as finding an invertible matrix $P$ such that $P^T\!\!AP$ is diagonal.

The second is for the linear operator generated by this matrix, finding an orthonormal basis consisting of eigenvectors of T. Which is the same as finding an orthogonal matrix $P$ (which consist of the normalised eigenvectors) such that $P^T \!\! AP$ is diagonal (which consists of the eigenvalues of $V$).

Have I got this right? So for a symmetric matrix on a Euclidean space there is two ways of diagonalising the matrix?

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1 Answer 1

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Right, and these have two different generalizations to nonsymmteric matrices.

It will be a little cleaner if I talk about Hermitian matrices, which are more general than symmetric ones. Every Hermitian matrix can be written as $$A=U D U^{\dagger} \quad (1)$$ where $D$ is diagonal with real entries, $U$ is unitary, and $\dagger$ is the conjugate transpose. Recall that the condition that $U$ is unitary means $U^{\dagger} = U^{-1}$, so we can also write $$A = U D U^{-1} \quad (2)$$.

Now, let $A$ be a general matrix with complex entries. Then the two formulas above have different generalizations. Formula $(1)$ becomes $$A = U D V^{\dagger}$$ with $U$ and $D$ unitary and $D$ real. This is the singular value decomposition, and every matrix has one.

Formula $(2)$ becomes $$A = S \Lambda S^{-1}.$$ where $S$ is an arbitrary complex matrix and $\Lambda$ is a diagonal matrix with complex entries. The entries of $\Lambda$ are the eigenvalues of $A$. Not quite every matrix has an eigendecomposition, but all but a measure zero subset do.

The eigenvalues and the singular values are, in general, different. Eigenvalues tend to be more useful when thinking about powers of $A$, and singular values tend to be more useful when thinking of $A$ as a quadratic form. The special thing about symmetric matrices is that the two coincide.

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