Eigen Values and Eigen Vectors Study Notes

1.2.     Echelon Form:

A matrix is in echelon form if only if

(i). Leading non-zero element in every row is behind leading non-zero element in previous row i.e. below the leading non-zero element in every row all the elements must be zero.

(ii). All the zero rows should be below all the non-zero rows.

1.2.1 Use of Echelon form:

This definition gives an alternate way of calculating the rank of larger matrices (larger than 3 × 3) more easily. The number of non-zero rows in the upper triangular matrix to get the rank of the matrix.

1.2.2 How to reduce a matrix into Echelon form?

To reduce a matrix to its echelon form, use gauss elimination method on the matrix and convert it into an upper triangular matrix, which will be in echelon form.

1.3. Elementary transformation of a matrix:

The following operations, three of which refer to rows and three to columns are known as elementary transformations:

(i). The interchange of any two rows (columns).

(ii). The multiplication of any row (column) by a non-zero number.

(iii). The addition of a constant multiple of the elements of any row (column) to the corresponding elements of any other row (column).

Note:

(1). Elementary transformations do not change either the order or rank of a matrix.

(2). While the value of the minors may get changed by the transformation I and II, their zero or non-zero character remains unaffected.

2.   Eigen Values, Eigen vectors and Cayley Hamilton Theorem:

2.1 Eigen Values:

Let A = [a_ij]_n×n be any n-rowed square matrix and  is a scalar. Then the matrix |A - λI | is called characteristic matrix of A, where I is the unit matrix of order n.

The  values of this characteristic equation are called eigen values of A and the set of eigenvalues of A is called the “spectrum of A”.

The corresponding non-zero solutions to X such that AX = λX, for different eigen values are called as the eigen vectors of A.

2.1.1 Properties of Eigen Values:

3. Eigen Vectors:

The corresponding non-zero solutions to X such that AX = λX, for different eigen values are called as the eigen vectors of A.

3.1 Properties of Eigen vectors:

(a). For each eigen value of a matrix there are infinitely many eigen vectors. If X is an eigen vector of a matrix A corresponding to the Eigen Value λ then KX is also an eigen vector of A for every non – zero value of K.

(b). Same Eigen vector cannot be obtained for two different eigen values of a matrix.

(c). Eigen vectors corresponding to the distinct eigen values are linearly independent.

(d). For the repeated eigen values, eigen vectors may or may not be linearly independent.

(e). The Eigen vectors of A and A^k are same.

(f). The eigen vectors of A and A^-1 are same.

(g). The Eigen vectors of A and A^T are NOT same.

(h). Eigen vectors of a symmetric matrix are Orthogonal.

4. Cayley Hamilton Theorem:

Every square matrix A satisfies its own characteristic equation A – λI = 0.

Example:

If λ² – 5λ + 6 =0 is the Characteristic equation of the matrix A, then according to Cayley Hamilton theorem:

A² – 5A +6I = 0

4.1 Applications of Cayley Hamilton theorem:

(a). It is used to find the higher powers of A such that A², A³, A⁴ etc.

(b). It can also be used to obtain the inverse of the Matrix.

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