

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 (GaussJordan) 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 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 is. 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 also has another special property that the eigenvectors are linearly independent. If we can find 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 are 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 are distinct (all the eigenvalues are simple), then matrix can be transformed into a diagonal matrix simply by formula. Modal matrix is formed by horizontal concatenation of the linearly independent eigenvectors. Since the matrix is symmetric, its modal matrix is orthogonal . The eigenvalues of lie on the main diagonal of. 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)? See also: Matrix Eigen Value & Eigen Vector, Similarity and Matrix Diagonalization, Matrix Power Rate this tutorial or give your comments about this tutorial Preferable reference for this tutorial is Teknomo, Kardi (2011) Linear Algebra tutorial. http:\\people.revoledu.com\kardi\ tutorial\LinearAlgebra\ 



