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Eigen Values & Eigen Vector

1-Lecture on Eigen Values
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The concept of Eigen Value and the method to find it out.
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If the eigen value of the matrix is 1, 2 and 3 then find the eigen value of following:

(a)    A-1

(b)    A+ 4A + I

(c)    Adj(A)

Lecture on Eigen Vectors
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Eigen vector of a matrix

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The eigen value of matrix \(A= \begin{bmatrix} 1 & 2\\ 0 & 2\\ \end{bmatrix}\)are written in the form \(\begin{bmatrix} 1\\ a\end{bmatrix}\)and \(\begin{bmatrix} 1\\ b \end{bmatrix}\) then, what will be the value of \(a\)  and  \(b\)?

Which one of the following is an eigen vector of the matrix \(\begin{bmatrix} 5 & 0 & 0 & 0\\ 0 & 5 & 5 & 0\\ 0 & 0 & 2 & 1\\ 0 & 0 & 3 & 1\end{bmatrix}\) ?

(a) \(\begin{bmatrix} 1\\ -2\\0\\0 \end{bmatrix}\)                                    (b)    \(\begin{bmatrix} 0\\ 0\\1\\0 \end{bmatrix}\)

(c)    \(\begin{bmatrix} 1\\0 \\0\\-2 \end{bmatrix}\)                                  (d)    \(\begin{bmatrix} 1\\ 0\\1\\0 \end{bmatrix}\)

For matrix \(A= \begin{bmatrix} 2 & -1 & 1\\ -1 & 2 & -1\\ 1 & -1 & 2\\ \end{bmatrix}\), find the value of   A6 - 6A5 + 9A4 - 2A3 - 12A2 + 23A - 9I

Example on Eigen Value

How many of the following matrices have an eigenvalue 1?

\(\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}\)\(\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}\)\(\begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix}\) and \(\begin{bmatrix} -1 & 0 \\ 1 & -1 \\ \end{bmatrix}\)

 

a) 1

b) 2 

c) 3

d) 4

For a matrix \(A\), and eigen value \(\lambda\),

 \(|A - \lambda I | = 0\) is the characteristic equation.

Lets look at the eigen values for each matrix given in the question 

(i) for \(\begin{bmatrix} 1 & 0 \\ 0 & 0 \\ \end{bmatrix}\) 

\(\begin{vmatrix} 1-\lambda & 0 \\ 0 & - \lambda\\ \end{vmatrix} = 0\)

\((1- \lambda)(- \lambda) = 0\)

⇒ \(\lambda = 0 \ \ and \ \ 1\)

 

(ii) For \(\begin{bmatrix} 0 & 1 \\ 0 & 0 \\ \end{bmatrix}\) ,

\(\begin{vmatrix} - \lambda & 1 \\ 0 & -\lambda \\ \end{vmatrix} = 0\)

⇒ \(\lambda^2 - 0 = 0 \)

\(\lambda = 0 \ \ and \ \ 0\)

(iii) for \(\begin{bmatrix} 1 & -1 \\ 1 & 1 \\ \end{bmatrix}\)

\(\begin{vmatrix} 1- \lambda & -1 \\ 1 & 1-\lambda \\ \end{vmatrix} = 0\)

⇒ \((1 - \lambda)^2 + 1 = 0\) 

\((1 - \lambda)^2 - (-1)^2 = 0\)

⇒ \((1 - \lambda -1) (1 - \lambda + 1) = 0\)

\((- \lambda)(2 - \lambda) = 0\)

⇒ \(\lambda = 0 \ \ and \ \ 2\)

 

(iv) for \(\begin{bmatrix} -1 & 0 \\ 1 & -1 \\ \end{bmatrix}\),

\(\begin{vmatrix}
-1 - \lambda & 0 \\
1 & -1 - \lambda
\end{vmatrix} = 0\)

\((-1 - \lambda)^2 - 1^2 = 0\)

⇒ \((-\lambda) (- \lambda - 2) = 0\)

⇒ \(\lambda = 0 \ \ and \ \ -2\)

Therefore, Ans is clearly (a) 1

 

3Comments
Some property of Eigen Values and Eigen vector
  1. The matrices A and AT have the same eigen values.                                                            we know that,
    (A-λI)=AT-λIT
    ∴ |(A-λI)T|=|AT-λI|                [i.e |AT|=|A|T]
    |A-λI|=|AT-λI|
    ∴ A & AT have same eigen values
  2. '0' is a characteristic root of a matrix if and only if the matrix is singular .i.e |A|=0
  3. The characteristic roots of a triangular matrix are just the diagonal elements of the matrix.
  4. If λ1,λ2...........λn are the eigen values of A ,then  kλ1,kλ2,kλ3......... are eigen values of kA.
  5. If  'A' is non singular, then eigen values of A-1 are the reciprocals of eigen values of 'A'.
  6. If 'λ' is a charactristicroot of matrix 'A' then (k+λ) is a characteristic root of matrix (A+kI).
  7. |A|=\prod ^{n}_{i=1}\lambda _{i}
  8. trace(A)=\sumλi  ,(i=1 to i=n)

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