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"Eigenvalue Solutions"
Power Method
λ =
x =

Matrix A = { { , , }, { , , }, { , , } }

[ Initial Values: {4,3,6},{3,7,1},{6,1,9} ]

Power Method

The power iteration is an eigenvalue algorithm. For any given real symmetric matrix A, the algorithm will generate an eigenvalue λ and a non zero eigenvector x, such that,

Ax = λ

The power iteration is probably the simplest method for finding the eigenvalue and eigenvector of a real symmetric matrix. However, it will find only a single eigenvalue (the one with the largest absolute value) and it may converge slowly.

Algorithm Creation

Suppose A is an [n x n] real symmetric matrix and v0 is any vector of length n.

We perform the following operations: (v-vector, A-matrix)

  • Multiply v0 by A.

  • Normalize the new vector Av0 to unit length.

  • Repeat the above process.

The above operations generate a series of vectors, v0, v1, v2,... . It turns out that these vectors converge to the eigenvector that corresponds to the eigenvalue with the largest absolute. The proof to confirm this theory is not shown.

The Power method is a method for finding the largest eigenvalue (in absolute value) and the corresponding eigenvector.

Running this set up produces the results shown above.

We believe the best way to show how our Power algorithm works is by running real-time and showing results ready to compare. The reader can try variations above by entering new values.

Testing the Eigenvalue Power Method

In order to test the Power method as defined above, a new TestPower() static method has been added and executed. Supporting code and methods are not shown.

           static void TestPower();
                 double[] x;
                 MatrixR A = new MatrixR(new double[,] { { t1, t2, t3 }, { t4, t5, t6 },
                  { t7, t8, t9 } });
                 VectorR x;
                 double lambda;
                 Eigenvalue.Power(A, 1e-5, out x, out lambda);
                 ListBox1.Items.Add(" " + lambda);
                 ListBox2.Items.Add(" " + x);

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