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Eigen Values and Eigen Vectors

Eigen Values

Let us consider a matrix \(A,\) a vector \(v\) and a scalar \(\lambda,\) if the matrix \(A\) satisfies the equation,

\[ Av=\lambda v \]

then, the values that \(\lambda\) attains are known as Eigen Values.

How to find Eigen Values?

To find eigen values, let us do some manipulation of the equation above,

\[ Av=\lambda v\newline \to Av=\lambda Iv\newline \to Av-\lambda Iv=0\newline \to v(A-\lambda I) = 0 \]

So now for non trivial solutions of the equation above 👆, i.e., \(v\neq0\) vector, we solve \(A-\lambda I=0.\)

The values of \(\lambda\) thus obtained are Eigen Values

Eigen Vectors

Now as we have learnt about Eigen Values we will now learn about Eigen Vectors.

Eigen vectors are basically the values of vector \(v\) corresponding to each eigenvalue \(\lambda,\)

Finding Eigen Vectors

After finding Eigen Values, put each value of \(\lambda\) in the equation \(Av=\lambda v\) or \((A-\lambda I)v=0\) and find the value of \(v\)

The values of \(v\) thus obtained are the eigen vectors to your matrix \(A\) corresponding to the eigen values \(\lambda.\)

Some Important Properties

  1. The sum of the eigen values is equal to the trace of the matrix \(A,\) i.e., \(\sum\lambda=tr(A).\)
  2. The product of the eigen values is equal to the determinant of the matrix \(A,\) i.e., \(\prod\lambda=det(A).\)
  3. A quick and easy way to find out the eigen values is by using the first two properties.
  4. Symmetric matrices have real eigenvalues.
  5. There may be some matrices for which there may be repeated eigen values.
  6. Power law: For a matrix \(A\) if there eigenvalues are \(\{\lambda_1,\lambda_2,...,\lambda_n\},\) then for \(A^k\) the eigenvalues will be \(\{\lambda_1^k, \lambda_2^k,...\lambda_n^k\}\)

Importance of Eigen Values and Eigen Vectors

If \(x\) is an eigen vector, then \(A\) either extends it or shrinks it by the scale of \(\lambda\)

This has been gracefully demonstrated by the professor in the lecture and if you prefer an animated version of the same, here is a great video by 3blue1brown