Virtual Labs

Eigen values, eigen vectors and diagonalization

Consider the vector space

M_{2 imes2}

over

\mathbb{R}

. Let Error in LaTeX 'A=egin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}': KaTeX parse error: Unexpected character: '' at position 3: A=̲egin{bmatrix} 2…. Then the characteristic polynomial and eigenvalues of

A

are

(\lambda + 2)^2

and

2

only Explanation

Explanation

(\lambda + 2)

and

1

only Explanation

Explanation

(\lambda - 2)^2

and

2

only Explanation

Explanation

(2 - \lambda)^2

and

-2

only Explanation

Explanation

Consider the vector space

M_{3 imes3}

over

\mathbb{R}

. Let Error in LaTeX 'A=egin{bmatrix} 1 & 0 & 0 \\ 0 & 1 & -1 \\ 0 & -1 & -1 \end{bmatrix}': KaTeX parse error: Unexpected character: '' at position 3: A=̲egin{bmatrix} 1…. Then the eigenvalues of

A

are

a: 1 only Explanation

Explanation

b: -1, 1 and 1 Explanation

Explanation

c: 1, -1 and 0 Explanation

Explanation

d: 0, 1 and 2 Explanation

Explanation

Let

A

be a

2 imes2

real matrix. Then

A

has

a: 2 distinct eigenvalues Explanation

Explanation

b: 1 eigenvalue only Explanation

Explanation

c: No eigenvalue Explanation

Explanation

d: Atmost 2 eigenvalue only Explanation

Explanation

Consider the vector space

M_{2 imes2}

over

\mathbb{R}

. Let

A

be matrix of order

2 imes2

which have eigenvalues

\alpha

and

\mu

, such that

\alpha<\mu

. Then

a: Error in LaTeX 'E_{\alpha} \cap E_{\mu} = arphi': KaTeX parse error: Unexpected character: '' at position 27: …\cap E_{\mu} = ̲arphi Explanation

Explanation

E_{\alpha} \subseteq E_{\mu}

Explanation

c: $x otin E_{\alpha} \cap E_{\mu}

, if

x eq 0$ Explanation

Explanation

d: $x otin E_{\alpha} \cap E_{\mu}

, if

x \eq 0$ Explanation

Explanation

Consider the vector space

M_{2 imes2}

over

\mathbb{R}

. Let Error in LaTeX 'A=egin{bmatrix} -1 & 0 \\ 2 & 0 \end{bmatrix}': KaTeX parse error: Unexpected character: '' at position 3: A=̲egin{bmatrix} -…. Then find the eigenvectors of

A

w.r.t. the eigenvalue

0

(-x, 2x)

, where

x \in \mathbb{R} \setminus \{0\}

Explanation

(0, x)

, where

x \in \mathbb{R}

Explanation

(-x, 0)

, where

x \in \mathbb{R}

Explanation

(-x, 2x)

, where

x \in \mathbb{R}

Explanation

Consider the vector space

M_{2 imes2}

over

\mathbb{R}

. Let Error in LaTeX 'A=egin{bmatrix} -1 & 0 \\ 2 & 0 \end{bmatrix}': KaTeX parse error: Unexpected character: '' at position 3: A=̲egin{bmatrix} -…. Then the eigen space of

A

w.r.t. the eigenvalue

-1

a: {

(1, -2)x

: where

x \in \mathbb{R}

} Explanation

Explanation

b: {

(0, x)

: where

x \in \mathbb{R} \setminus \{0\}

} Explanation

Explanation

c: {

(1, 0)x

: where

x \in \mathbb{R}

} Explanation

Explanation

\mathbb{R}^2

Explanation

Let

1

and

2

be eigenvalues of a matrix

A

. Let

x

be an eigenvector corresponding to the eigenvalue

1

and

y

be an eigenvector corresponding to the eigenvalue

2

. Then

3

is an eigenvalue and

x+y

is an eigenvector corresponding to the eigenvalue

3

Explanation

x+y

is an eigenvector corresponding to both the eigenvalues

1

and

2

Explanation

2x

is an eigenvector corresponding to the eigenvalue

1

and

y+2

is not an eigenvector corresponding to the eigenvalue

2

Explanation

2x

is not an eigenvector corresponding to the eigenvalue

1

and

y+2

is an eigenvector corresponding to the eigenvalue

2

Explanation