eigenvalues and eigenvectors theorem eigenvalues of hermitian matrix any hermitian matrix has real eigenvalues the corresponding matrix of normalized eigenvectors is unitary matrix prooflet be hermitian matrix by theorem there exists unitary matrix such that - au twhere is upper triangular it follows that the adjoint ( - au)tis lower triangular however( - au) - ausince aa and uu- hencet and tmust be the samewhich can only be the case if is real diagonal matrix since au duthe diagonal elements are exactly the eigenvalues and the corresponding eigenvectors are the columns of in particularthe eigenvalues of real symmetric matrix are real we can now repeat the proof of theorem with real eigenvalues and eigenvectorsso that there exists an orthogonal matrix such that - aq qaq the eigenvectors can be chosen as the columns of qwhich form an orthonormal basis this proves the following theorem theorem real symmetric matrices are orthogonally diagonizable any real symmetric matrix can be written as qdqwhere is the diagonal matrix of (realeigenvalues and is an orthogonal matrix whose columns are eigenvectors of example (real symmetric matrices and ellipsesas we have seenlinear transformations map circles into ellipses we can use the above theory for real symmetric matrices to identify the principal axes considerfor examplethe transformation with matrix [ - / - in ( point on the unit circle is mapped to point ax since for such points kxk xx we have that satisfies ( - ) - which gives the equation for the ellipse let be the orthogonal matrix of eigenvectors of the symmetric matrix ( - ) - (aa)- so (aa)- for some diagonal matrix taking the inverse on both sides of the previous equationwe have qaaq - which shows that is also the matrix of eigenvectors of aathese eigenvectors point precisely in the direction of the principal axesas shown in figure it turns outsee section that the square roots of the eigenvalues of aahere approximately and correspond to the sizes of the principal axes of the ellipseas illustrated in figure