Find the eigenvalues and eigenvectors of A geometrically over the real numbers ℝ. (If an eigenvalue does not exist, enter DNE. If an eigenvector does not exist, enter DNE in any single blank.) A = 0 1 1 0 (reflection in the line y = x) λ1 = has eigenspace span (smaller λ-value) λ2 = has eigenspace span (larger λ-value)

We want to find the eigenvalues and eigenvectors of the given matrix.

The eigenvalues are 1 and -1 and the correspondent eigenvectors are:

 and  

We have the matrix:

To find the eigenvalues λ, we need to solve:

This gives:

Then the eigenvalues are 1 and -1

Now let's get the eigenvectors.

For λ = 1 we must solve:

Then a = -b

This eigenvector is:

(the factor is for normalization)

We do the same thing for the other eigenvalue:

Now we have a = b

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∴The eigen values are -1,1.

The eigen vector for 1 is     .

The eigen vector for = - 1 is     .

Step-by-step explanation:

Given matrix is

To find the eigen values of the given matrix, we set

∴The eigen values are -1,1.

For

Let the eigen vector for is

∴

let

The eigen vector for 1 is     .

For

Let the eigen vector for is

∴

and

From the above equations, we get

Let

Then,

The eigen vector for = - 1 is     .

x=-7

step-by-step explanation:

x+3=-10

-10+3=-7

x=-7

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step-by-step explanation: