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Matrix Size & Validation
Vector Algebra
What is Vector?
Vector Norm
Unit Vector
Vector Addition
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Vector Scalar Multiple
Vector Multiplication
Vector Inner Product
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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
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Is Equal Matrix?
Matrix Transpose
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Matrix Multiplication
Matrix Scalar Multiple
Hadamard Product
Horizontal Concatenation
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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

Matrix Eigen Value & Eigen Vector for Symmetric Matrix

By Kardi Teknomo, PhD.
LinearAlgebra

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Symmetric matrix is a special type of square matrix that often come in many applications such as covariance matrix, correlation matrix and distance matrix. In this page, you will learn more about the eigenvalues and eigenvectors of symmetric matrix.

Symmetric matrix is matrix where the transpose is equal to itself, that isMatrix Eigen Value & Eigen Vector for Symmetric Matrix. Symmetric matrix has a special property that the eigenvalues of symmetric matrix is always real number (remember that eigenvalues can be complex number). This greatly simplifies many applications because you do not need to handle complex number arithmetic.

Furthermore, symmetric matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrixalso has another special property that the eigenvectors are linearly independent. If we can find Matrix Eigen Value & Eigen Vector for Symmetric Matrix linearly independent eigenvectors, these eigenvectors are the right candidate for basis vectors to create new space indicated by new coordinate system. Not only that, there is another property of symmetric matrix that if the eigenvalues of symmetric matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrixare distinct (all eigenvalues are different with no multiple eigenvalues), then the eigenvectors that belong to distinct eigenvalues are orthogonal. This is a very nice property because using symmetric matrix you will get basis vectors, and these basis vectors are guaranteed to be perpendicular to each other. Some applications such as Principal Component Analysis (PCA) and Multi Dimensional Scaling (MDS) rely heavily on these properties.

Lastly but not least, if you can find that all eigenvalues of a symmetric matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrixare distinct (all the eigenvalues are simple), then matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrix can be transformed into a diagonal matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrixsimply by formulaMatrix Eigen Value & Eigen Vector for Symmetric Matrix. Modal matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrix is formed by horizontal concatenation of the Matrix Eigen Value & Eigen Vector for Symmetric Matrixlinearly independent eigenvectorsMatrix Eigen Value & Eigen Vector for Symmetric Matrix. Since the matrix Matrix Eigen Value & Eigen Vector for Symmetric Matrixis symmetric, its modal matrix is orthogonal Matrix Eigen Value & Eigen Vector for Symmetric Matrix. The eigenvalues of Matrix Eigen Value & Eigen Vector for Symmetric Matrix lie on the main diagonal ofMatrix Eigen Value & Eigen Vector for Symmetric Matrix. Diagonal matrix simplified much computation because it behaves almost similar to scalar. For example, you can take any function such as power or exponent and even derivative only to the diagonal elements.

The interactive program below helps you to find eigenvalues and eigenvectors of a symmetric matrix. The program is using Jacobi method to compute symmetric Eigen system. When you click “Random Example” button, the program will generate random symmetric matrix. Practically, this Random Example gives you unlimited examples for you to investigate the behavior of eigenvalues and eigenvectors of a symmetric matrix. For instance, you may check that the eigenvectors are orthonormal (the norm of each row and each column is one). Can you find any example that the eigenvalues are multiple to each other (not all distinct)?


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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, Similarity and Matrix Diagonalization, Matrix Power

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Preferable reference for this tutorial is

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

 

 
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