Linear Algebra Examples

Find the Eigenvectors/Eigenspace [[5,0],[1,5]]
Step 1
Find the eigenvalues.
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Step 1.1
Set up the formula to find the characteristic equation .
Step 1.2
The identity matrix or unit matrix of size is the square matrix with ones on the main diagonal and zeros elsewhere.
Step 1.3
Substitute the known values into .
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Step 1.3.1
Substitute for .
Step 1.3.2
Substitute for .
Step 1.4
Simplify.
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Step 1.4.1
Simplify each term.
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Step 1.4.1.1
Multiply by each element of the matrix.
Step 1.4.1.2
Simplify each element in the matrix.
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Step 1.4.1.2.1
Multiply by .
Step 1.4.1.2.2
Multiply .
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Step 1.4.1.2.2.1
Multiply by .
Step 1.4.1.2.2.2
Multiply by .
Step 1.4.1.2.3
Multiply .
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Step 1.4.1.2.3.1
Multiply by .
Step 1.4.1.2.3.2
Multiply by .
Step 1.4.1.2.4
Multiply by .
Step 1.4.2
Add the corresponding elements.
Step 1.4.3
Simplify each element.
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Step 1.4.3.1
Add and .
Step 1.4.3.2
Add and .
Step 1.5
Find the determinant.
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Step 1.5.1
The determinant of a matrix can be found using the formula .
Step 1.5.2
Simplify the determinant.
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Step 1.5.2.1
Expand using the FOIL Method.
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Step 1.5.2.1.1
Apply the distributive property.
Step 1.5.2.1.2
Apply the distributive property.
Step 1.5.2.1.3
Apply the distributive property.
Step 1.5.2.2
Simplify and combine like terms.
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Step 1.5.2.2.1
Simplify each term.
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Step 1.5.2.2.1.1
Multiply by .
Step 1.5.2.2.1.2
Multiply by .
Step 1.5.2.2.1.3
Multiply by .
Step 1.5.2.2.1.4
Rewrite using the commutative property of multiplication.
Step 1.5.2.2.1.5
Multiply by by adding the exponents.
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Step 1.5.2.2.1.5.1
Move .
Step 1.5.2.2.1.5.2
Multiply by .
Step 1.5.2.2.1.6
Multiply by .
Step 1.5.2.2.1.7
Multiply by .
Step 1.5.2.2.2
Subtract from .
Step 1.5.2.3
Subtract from .
Step 1.5.2.4
Move .
Step 1.5.2.5
Reorder and .
Step 1.6
Set the characteristic polynomial equal to to find the eigenvalues .
Step 1.7
Solve for .
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Step 1.7.1
Factor using the perfect square rule.
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Step 1.7.1.1
Rewrite as .
Step 1.7.1.2
Check that the middle term is two times the product of the numbers being squared in the first term and third term.
Step 1.7.1.3
Rewrite the polynomial.
Step 1.7.1.4
Factor using the perfect square trinomial rule , where and .
Step 1.7.2
Set the equal to .
Step 1.7.3
Add to both sides of the equation.
Step 2
The eigenvector is equal to the null space of the matrix minus the eigenvalue times the identity matrix where is the null space and is the identity matrix.
Step 3
Find the eigenvector using the eigenvalue .
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Step 3.1
Substitute the known values into the formula.
Step 3.2
Simplify.
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Step 3.2.1
Simplify each term.
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Step 3.2.1.1
Multiply by each element of the matrix.
Step 3.2.1.2
Simplify each element in the matrix.
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Step 3.2.1.2.1
Multiply by .
Step 3.2.1.2.2
Multiply by .
Step 3.2.1.2.3
Multiply by .
Step 3.2.1.2.4
Multiply by .
Step 3.2.2
Add the corresponding elements.
Step 3.2.3
Simplify each element.
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Step 3.2.3.1
Subtract from .
Step 3.2.3.2
Add and .
Step 3.2.3.3
Add and .
Step 3.2.3.4
Subtract from .
Step 3.3
Find the null space when .
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Step 3.3.1
Write as an augmented matrix for .
Step 3.3.2
Find the reduced row echelon form.
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Step 3.3.2.1
Swap with to put a nonzero entry at .
Step 3.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 3.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 3.3.5
Write the solution as a linear combination of vectors.
Step 3.3.6
Write as a solution set.
Step 3.3.7
The solution is the set of vectors created from the free variables of the system.
Step 4
The eigenspace of is the list of the vector space for each eigenvalue.