linear algebra Linear algebra is the branch of mathematics concerning linear equations such as :a_1x_1+\cdots +a_nx_n=b, linear maps such as :(x_1, \ldots, x_n) \mapsto a_1x_1+\cdots +a_nx_n, and their representations in vector spaces and through matrix (mathemat ...

, an orthogonal diagonalization of a

normal matrix In mathematics, a complex square matrix is normal if it commutes with its conjugate transpose : :A \text \iff A^*A = AA^* . The concept of normal matrices can be extended to normal operators on infinite-dimensional normed spaces and to nor ...

(e.g. a

symmetric matrix In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with ...

) is a diagonalization by means of an

orthogonal In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendic ...

change of coordinates. The following is an orthogonal diagonalization algorithm that diagonalizes a

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two (" form" is another name for a homogeneous polynomial). For example, 4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong t ...

''q''(''x'') on

\mathbb

^''n'' by means of an orthogonal change of coordinates ''X'' = ''PY''. Seymour Lipschutz ''3000 Solved Problems in Linear Algebra.'' * Step 1: find the symmetric matrix ''A'' which represents ''q'' and find its

characteristic polynomial In linear algebra, the characteristic polynomial of a square matrix is a polynomial which is invariant under matrix similarity and has the eigenvalues as roots. It has the determinant and the trace of the matrix among its coefficients. The ...

\Delta(t).

* Step 2: find the

eigenvalue In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by a ...

s of ''A'' which are the

roots A root is the part of a plant, generally underground, that anchors the plant body, and absorbs and stores water and nutrients. Root or roots may also refer to: Art, entertainment, and media * ''The Root'' (magazine), an online magazine focusin ...

\Delta(t)

. * Step 3: for each eigenvalue

\lambda

of ''A'' from step 2, find an

orthogonal basis In mathematics, particularly linear algebra, an orthogonal basis for an inner product space V is a basis for V whose vectors are mutually orthogonal. If the vectors of an orthogonal basis are normalized, the resulting basis is an ''orthonormal b ...

of its

eigenspace In linear algebra, an eigenvector ( ) or characteristic vector is a Vector (mathematics and physics), vector that has its direction (geometry), direction unchanged (or reversed) by a given linear map, linear transformation. More precisely, an e ...

. * Step 4: normalize all eigenvectors in step 3 which then form an

orthonormal basis In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vec ...

\mathbb

^''n''. * Step 5: let ''P'' be the

matrix Matrix (: matrices or matrixes) or MATRIX may refer to: Science and mathematics * Matrix (mathematics), a rectangular array of numbers, symbols or expressions * Matrix (logic), part of a formula in prenex normal form * Matrix (biology), the m ...

whose columns are the normalized

eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...

s in step 4. Then ''X'' = ''PY'' is the required orthogonal change of coordinates, and the diagonal entries of

P^T A P

will be the eigenvalues

\lambda_1,\dots,\lambda_n

which correspond to the columns of ''P''.

References

* Maxime Bôcher (with E.P.R. DuVal)(1907) ''Introduction to Higher Algebra''
§ 45 Reduction of a quadratic form to a sum of squares
via

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Linear algebra {{linear-algebra-stub