Orthogonal diagonalization

Last updated April 30, 2019

In linear algebra, an orthogonal diagonalization of a symmetric matrix is a diagonalization by means of an orthogonal change of coordinates.^[1]

Linear algebra is the branch of mathematics concerning linear equations such as

Matrix (mathematics) Two-dimensional array of numbers with specific operations

In mathematics, a matrix is a rectangular array of numbers, symbols, or expressions, arranged in rows and columns. For example, the dimensions of the matrix below are 2 × 3, because there are two rows and three columns:

In linear algebra, a square matrix $is called diagonalizable or nondefective if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix such that is a diagonal matrix. If is a finite-dimensional vector space, then a linear map is called diagonalizable if there exists an ordered basis of with respect to which is represented by a diagonal matrix. Diagonalization is the process of finding a corresponding diagonal matrix for a diagonalizable matrix or linear map. A square matrix that is not diagonalizable is called defective.$

The following is an orthogonal diagonalization algorithm that diagonalizes a quadratic form q(x) on Rⁿ by means of an orthogonal change of coordinates X = PY.^[2]

In mathematics, a quadratic form is a polynomial with terms all of degree two. For example,

Step 1: find the symmetric matrix A which represents q and find its characteristic polynomial $\Delta (t).$
Step 2: find the eigenvalues of A which are the roots of $\Delta (t)$ .
Step 3: for each eigenvalues $\lambda$ of A in step 2, find an orthogonal basis of its eigenspace.
Step 4: normalize all eigenvectors in step 3 which then form an orthonormal basis of Rⁿ.
Step 5: let P be the matrix whose columns are the normalized eigenvectors in step 4.

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,

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 as coefficients. The characteristic polynomial of an endomorphism of vector spaces of finite dimension is the characteristic polynomial of the matrix of the endomorphism over any base; it does not depend on the choice of a basis. The characteristic equation is the equation obtained by equating to zero the characteristic polynomial.

Root system Geometric arrangements of points, foundational to Lie theory

In mathematics, a root system is a configuration of vectors in a Euclidean space satisfying certain geometrical properties. The concept is fundamental in the theory of Lie groups and Lie algebras, especially the classification and representation theory of semisimple Lie algebras. Since Lie groups and Lie algebras have become important in many parts of mathematics during the twentieth century, the apparently special nature of root systems belies the number of areas in which they are applied. Further, the classification scheme for root systems, by Dynkin diagrams, occurs in parts of mathematics with no overt connection to Lie theory. Finally, root systems are important for their own sake, as in spectral graph theory.

The X=PY is the required orthogonal change of coordinates, and the diagonal entries of $P^{T}AP$ will be the eigenvalues $\lambda _{1},\dots ,\lambda _{n}$ which correspond to the columns of P.

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In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it. More formally, if $T$ is a linear transformation from a vector space $V$ over a field $F$ into itself and $v$ is a vector in $V$ that is not the zero vector, then $v$ is an eigenvector of $T$ if $T (v)$ is a scalar multiple of $v$ . This condition can be written as the equation

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References

↑ Poole, D. (2010). Linear Algebra: A Modern Introduction (in Dutch). Cengage Learning. p. 411. ISBN 978-0-538-73545-2 . Retrieved 12 November 2018.
↑ Seymour Lipschutz 3000 Solved Problems in Linear Algebra.

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

Maxime Bôcher 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. Bôcher's theorem, Bôcher's equation, and the Bôcher Memorial Prize are named after him.

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

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[Poole_2010_p._411-1] Poole, D. (2010). Linear Algebra: A Modern Introduction (in Dutch). Cengage Learning. p. 411. ISBN 978-0-538-73545-2 . Retrieved 12 November 2018.

[2] Seymour Lipschutz 3000 Solved Problems in Linear Algebra.