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 matrices. ...

, an orthogonal diagonalization of a symmetric

matrix Matrix most commonly refers to: * ''The Matrix'' (franchise), an American media franchise ** ''The Matrix'', a 1999 science-fiction action film ** "The Matrix", a fictional setting, a virtual reality environment, within ''The Matrix'' (franchis ...

is a diagonalization by means of an

orthogonal In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in ...

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 to a ...

''q''(''x'') on R^''n'' by means of an orthogonal change of coordinates ''X'' = ''PY''.

Seymour Lipschutz Seymour Saul Lipschutz (born 1931 died March 2018) was an author of technical books on pure mathematics and probability, including a collection of Schaum's Outlines. Lipschutz received his Ph.D. in 1960 from New York University's Courant Institute ...

''3000 Solved Problems in Linear Algebra.'' * Step 1: find the

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 re ...

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 chara ...

\Delta (t).

* Step 2: find the

eigenvalues In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

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 focusing ...

\Delta (t)

. * Step 3: for each eigenvalue

\lambda

of A from step 2, find an orthogonal basis of its

eigenspace In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

. * Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of R^''n''. * Step 5: let P be the matrix whose columns are the normalized

eigenvector In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denoted b ...

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

P^T AP

will be the eigenvalues

\lambda_ ,\dots ,\lambda_

which correspond to the columns of P.

References

Maxime Bôcher Maxime Bôcher (August 28, 1867 – September 12, 1918) was an American mathematician who published about 100 papers on differential equations, series, and algebra. He also wrote elementary texts such as ''Trigonometry'' and ''Analytic Geometry''. ...

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

HathiTrust HathiTrust Digital Library is a large-scale collaborative repository of digital content from research libraries including content digitized via Google Books and the Internet Archive digitization initiatives, as well as content digitized locally ...

Linear algebra {{linear-algebra-stub