Kardi Teknomo
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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
Vector Subtraction
Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
Vector Outer Product
Vector Cross Product
Vector Triple Cross Product
Vector Triple Dot Product
Scalar Triple Product
Orthogonal & Orthonormal Vector
Cos Angle of Vectors
Scalar and Vector Projection
Matrix Algebra
What is a matrix?
Special Matrices
Matrix One
Null Matrix
Matrix Diagonal Is Diagonal Matrix?
Identity Matrix
Matrix Determinant
Matrix Sum
Matrix Trace
Matrix Basic Operation
Is Equal Matrix?
Matrix Transpose
Matrix Addition
Matrix Subtraction
Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
Vertical Concatenation
Elementary Row Operations
Matrix RREF
Finding inverse using RREF (Gauss-Jordan)
Finding Matrix Rank using RREF
Matrix Inverse
Is Singular Matrix?
Linear Transformation
Matrix Generalized Inverse
Solving System of Linear Equations
Linear combination, Span & Basis Vector
Linearly Dependent & Linearly Independent
Change of basis
Matrix Rank
Matrix Range
Matrix Nullity & Null Space
Eigen System
Matrix Eigen Value & Eigen Vector
Symmetric Matrix
Matrix Eigen Value & Eigen Vector for Symmetric Matrix
Similarity Transformation and Matrix Diagonalization
Matrix Power
Orthogonal Matrix
Spectral Decomposition
Singular Value Decomposition
Resources on Linear Algebra

Spectral Decomposition

By Kardi Teknomo, PhD.

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Diagonalization of a real symmetric matrix is also called spectral decomposition, or Schur Decomposition. Given a square symmetric matrixSpectral Decomposition, the matrix can be factorized into two matricesSpectral Decomposition andSpectral Decomposition. Matrix Spectral Decompositionis an orthogonal matrixSpectral Decomposition. MatrixSpectral Decomposition is a diagonal matrix. Spectral decomposition is matrix factorization because we can multiply the matrices to get back the original matrixSpectral Decomposition.

The interactive program below yield three matricesSpectral Decomposition,Spectral Decomposition and matrix Spectral Decompositionand also gives you feedback onSpectral Decomposition. Random example will generate random symmetric matrix.

Yes, this program is a free educational program!! Please don't forget to tell your friends and teacher about this awesome program!

See also: Matrix Eigen Value & Eigen Vector for Symmetric Matrix, Similarity and Matrix Diagonalization, Symmetric Matrix, Singular Value Decomposition

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This tutorial is copyrighted.

Preferable reference for this tutorial is

Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\


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