Matrices Eigenvalues

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Let A = [aij] be a real square matrix of order n. If there exists a non-zero column vector x and a scalar lambda , such that AX = lambda X, then lambda is called an Eigenvalue of A and X is called the Eigen vector corresponding to the Eigen Value lambda.

matrices eigenvalues-Characteristic Equation of a Matrix:

Let A=a=[aij] be a square matrix of order n. The linear transformation AX = Y transforms a column vector X into another column vector Y with respect to A.

AX = [[a11,a12,a1n],[a21,a22,a2n],[an1,an2,an n]] [[x1],[x2],[xn]] = [[y1],[y2],[y3]] = Y.

If AX = lambda X, for some scalar lambda , then the vector X is transformed into lambda x.

AX = lambda X.

(A - lambda I) = 0.

{ [[a11,a12...,a1n],[a21,a22...,a2n],[an1,an2...,an n]] – lambda [[1,0...,0],[0,1...,0],[0,0...,1]] } [[x1],[x2],[xn]] = [[0],[0],[0]] .

(a11 - lambda )x1 + a12x2 + ….. + a1nxn = 0.

a21x1 + (a22 - lambda )x2 + ….. + a2nxn = 0.

an1x1 + an2x2 + ….. + (an n - lambda )xn = 0. …..........(1).

The condition for the system (1) to have a non-trivial solution (non-zero solution) ...
... is,

[[a11-lambda,a12...,a1n],[a21,a22- lambda...,a2n],[an1,an2...,an n- lambda]] - = 0 ....(2).

|A - lambda I| = 0.

The equation |A - lambda I| = 0, or the equation (2) is called the characteristic equation of A.

matrices eigenvalues-Example:

Find the eigenvalue and eigen vectors of the matrix [[4,3],[3,2]].

Solution:

the given matrix, A = [[4,3],[3,2]] .

The characteristic equation of A is |A - lambda I| = 0.

|A - lambda I| = 0.

[[4-lambda,1],[3,2-lambda]] = 0.

(4- lambda )(2- lambda ) - 3 = 0.

lambda 2– 6 lambda + 5 = 0.

( lambda - 5)( lambda -1) = 0.

lambda = 1,5.

the eigen values are lambda = 1,5.

for the matrix A , the equation (A - lambda I)X = 0 becomes,

(4- lambda )x1 + x2 = 0.

3x1 + (2- lambda )x2 = 0. …...........(1).

Case:1

when lambda = 1, (1) becomes

3x1 + x2 = 0.

3x1 = -x2.

X1/-1 = x2/3.

Hence the Corresponding eigen vector is X1 = [[-1],[3]].

Case:2

when lambda = 5, (1) becomes

-x1 + x2 = 0.

3x1 – 3x2 = 0.

-x1 + x2 = 0.

x1/1 = x2/1.

Hence the corresponding eigen vector is X2 = [[1],[1]].

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The properties of symmetric matrices represents the matrices which have the square matrices ( having same row and same column) and the transpose of this matrix give the same matrix in dimension and also the elements. For symmetric property, consider the matrix A as A = [[1,2],[2,3]] having the dimension 2 x 2 is the symmetric matrix when A T = [[1,2],[2,3]] which is also the symmetric matrices appeared here.

Explanation about properties of symmetric matrices

If the matrices A and B are symmetric in n x n matrices, then

(A + B) T = AT + BT = (A + B) So (A + B) is symmetric.

If the matrices A and B are symmetric in n x n matrices, then

(A B) T = BT AT = BA != AB So AB is not symmetric.

If C is any n x n matrix then B = CTC is symmetric:

(CTC)T = CT (CT )T = CTC

If D is a diagonal matrix, then D is symmetric

Study about the symmetric matrices for the two matrices [[1,2,3],[2,4,5],[3,5,6]] and [[11,12,13],[12,14,15],[13,15,16]]

Solution:

show (D + E) T = DT + ET = (D + E)

Let D be [[D_11,D_12 ,D_13 ],[D_21 ,D_22 ,D_23 ],[D_31 ,D_32 ,D_33 ]] and and E be [[E_11,E_12 ,E_13 ],[E_21 ,E_22 ,E_23 ],[E_31 ,E_32 ,E_33 ]]

Let D be [[1,2,3],[2,4,5],[3,5,6]] and and E be [[11,12,13],[12,14,15],[13,15,16]]

D + E = [[D_11 + E_11,D_12 + E_12,D_13 + E_13],[D_21 + E_21,D_22 + E_22,D_23 + E_23],[D_31 + E_31,D_32 + E_32,D_33 + E_33]]

D + E = [[1+11,2+12,3+13],[2+12,4+14,5+15],[3+13,5+15,6+16]]

D + E = [[12,14,16],[14,18,20],[16,20,22]] -------------------- (1)

(D + E) T = [[12,14,16],[14,18,20],[16,20,22]] -------------------- (2)

DT = [[1,2,3],[2,4,5],[3,5,6]]

ET = [[11,12,13],[12,14,15],[13,15,16]]

DT + ET = [[1+11,2+12,3+13],[2+12,4+14,5+15],[3+13,5+15,6+16]]

DT + ET = [[12,14,16],[14,18,20],[16,20,22]] -------------------- (3)

From (1), (2) and (3)

We have

(D + E) T = [[12,14,16],[14,18,20],[16,20,22]] = DT + ET = [[12,14,16],[14,18,20],[16,20,22]] = (D + E)

Therefore the property is proved.

problems to explain "properties of symmetric matrices"

If D is a diagonal matrix, then D is symmetric

Consider the diagonal matrix D = [[11,0,0],[0,11,0],[0,0,11]] show it is symmetric

Solution:

Here given D = [[11,0,0],[0,11,0],[0,0,11]]

then DT = [[11,0,0],[0,11,0],[0,0,11]]

Therefore D = DT = [[11,0,0],[0,11,0],[0,0,11]] is the symmetric matrix.

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