Eigen values and Eigenvectors of Matrix
Let A be a n×n matrix, λ is an Eigen value for A if there exists a vector u≠0 such that
Au = λu
If such a vector u exists, it is said to be an eigenvector associated with the Eigen value λ.
Suppose that λ is an Eigen value for A.
Then there exists a non-zero vector such that Au = λu or, equivalently, such that (A - λIn)u = 0, where In is the n×n identity matrix.
Since (A - λIn)u = 0 has a non-trivial solution u, the matrix (A - λIn) is not invertible, i.e. |A - λIn| = 0.
Note that |A - λIn| is a polynomial in λ, so you get the eigenvalues of A by finding the roots of |A - λIn|
Say λ0 is such an eigenvalue. In order to find the eigenvectors associated with λ0, you have to solve the system Au = λ0u for x1, …, xn.
Example
To find the Eigen values and Eigen vectors of two by two matrices
Let
To find the Eigen values of A, form the matrix A - λI2, find its determinant |A - λI2| and solve the equation |A - λI2| = 0:
Now,
|A -λI2| = (-3-λ)(1-λ)-(2)(-2) = λ2+2λ+1,
so that the eigenvalues of A are the roots of λ2 + 2λ + 1 = (λ+1)2, i.e. A has a repeated eigenvalue: λ0 = -1.
Now we have to solve the system Au = λ0u. Here λ0 = -1, so that
Au = λ0u ⇔ Au = -u,
which yields
or, equivalently,
A vector is therefore an eigenvector associated with the eigen value λ0 = -1 if and only if its coordinates satisfy
,
i.e. if and only if x1 = x2.
Hence the eigenvectors associated with the eigen value λ0 = -1 are of the form , where α is a real number.
Diagonalisation of Matrix
The elements of a square matrix which lie in the sane row and same column position are said to form the diagonal of a square matrix. i.e. in a square matrix the elements are said to form the diagonal of a matrix .
For example in the matrix the elements 1, 0 and 4 are the (1,1), (2,2) and (3,3) elements and hence form the diagonal of the square matrix A.
A square matrix is said to be a diagonal matrix if its non-diagonal elements are all zeros, where diagonal elements may or may not zero.
i.e. If is a square matrix of order ‘n’ then it will be a diagonal matrix if aij = 0, ∀ i ≠ j, where 1 ≤ i, j ≤ n.
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