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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 |
Matrix Power A repeat multiplication of a square matrix When the power When the power
Example: Then, matrix power can be computed through repeated matrix multiplication 4 times
However, we have something else that matrix multiplication cannot perform: when matrix power Example: See also: Matrix Eigen Value & Eigen Vector, Similarity and Matrix Diagonalization 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\ |
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with itself 
times produces matrix power
, thus
is large, computation of matrix multiplication takes a long time and a more efficient method is to transform matrix 
into a similar diagonal matrix 
using 
. Matrix 
(is called modal matrix) is formed by 
. Matrix 
can be obtained back from diagonal matrix using
. The computation of matrix power is simplified by taking the power to the scalar diagonal elements. Then multiply the diagonal power with the modal matrix and its inverse
.
is not integer the computation of matrix power is possible only through similarity transformation into 
has an 
we can obtain diagonal matrix
. Through modal matrix (augmented eigenvectors)
. The inverse modal matrix is
.
. The same result can be obtained using diagonal matrix
. Then we multiply with modal matrix and its inverse, require only three matrix multiplications
.
is large.
is not integer.
. We multiply back to obtain the matrix power
.