Linear Algebra Examples

Find the Eigenvectors/Eigenspace [[5,0,1],[-3,1,0],[-3,0,1]]
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 .
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Step 1.4.1.2.4.1
Multiply by .
Step 1.4.1.2.4.2
Multiply by .
Step 1.4.1.2.5
Multiply by .
Step 1.4.1.2.6
Multiply .
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Step 1.4.1.2.6.1
Multiply by .
Step 1.4.1.2.6.2
Multiply by .
Step 1.4.1.2.7
Multiply .
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Step 1.4.1.2.7.1
Multiply by .
Step 1.4.1.2.7.2
Multiply by .
Step 1.4.1.2.8
Multiply .
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Step 1.4.1.2.8.1
Multiply by .
Step 1.4.1.2.8.2
Multiply by .
Step 1.4.1.2.9
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 1.4.3.3
Add and .
Step 1.4.3.4
Add and .
Step 1.4.3.5
Add and .
Step 1.4.3.6
Add and .
Step 1.5
Find the determinant.
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Step 1.5.1
Choose the row or column with the most elements. If there are no elements choose any row or column. Multiply every element in column by its cofactor and add.
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Step 1.5.1.1
Consider the corresponding sign chart.
Step 1.5.1.2
The cofactor is the minor with the sign changed if the indices match a position on the sign chart.
Step 1.5.1.3
The minor for is the determinant with row and column deleted.
Step 1.5.1.4
Multiply element by its cofactor.
Step 1.5.1.5
The minor for is the determinant with row and column deleted.
Step 1.5.1.6
Multiply element by its cofactor.
Step 1.5.1.7
The minor for is the determinant with row and column deleted.
Step 1.5.1.8
Multiply element by its cofactor.
Step 1.5.1.9
Add the terms together.
Step 1.5.2
Multiply by .
Step 1.5.3
Multiply by .
Step 1.5.4
Evaluate .
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Step 1.5.4.1
The determinant of a matrix can be found using the formula .
Step 1.5.4.2
Simplify the determinant.
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Step 1.5.4.2.1
Simplify each term.
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Step 1.5.4.2.1.1
Expand using the FOIL Method.
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Step 1.5.4.2.1.1.1
Apply the distributive property.
Step 1.5.4.2.1.1.2
Apply the distributive property.
Step 1.5.4.2.1.1.3
Apply the distributive property.
Step 1.5.4.2.1.2
Simplify and combine like terms.
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Step 1.5.4.2.1.2.1
Simplify each term.
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Step 1.5.4.2.1.2.1.1
Multiply by .
Step 1.5.4.2.1.2.1.2
Multiply by .
Step 1.5.4.2.1.2.1.3
Multiply by .
Step 1.5.4.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 1.5.4.2.1.2.1.5
Multiply by by adding the exponents.
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Step 1.5.4.2.1.2.1.5.1
Move .
Step 1.5.4.2.1.2.1.5.2
Multiply by .
Step 1.5.4.2.1.2.1.6
Multiply by .
Step 1.5.4.2.1.2.1.7
Multiply by .
Step 1.5.4.2.1.2.2
Subtract from .
Step 1.5.4.2.1.3
Multiply .
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Step 1.5.4.2.1.3.1
Multiply by .
Step 1.5.4.2.1.3.2
Multiply by .
Step 1.5.4.2.2
Add and .
Step 1.5.4.2.3
Reorder and .
Step 1.5.5
Simplify the determinant.
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Step 1.5.5.1
Combine the opposite terms in .
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Step 1.5.5.1.1
Add and .
Step 1.5.5.1.2
Add and .
Step 1.5.5.2
Expand by multiplying each term in the first expression by each term in the second expression.
Step 1.5.5.3
Simplify each term.
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Step 1.5.5.3.1
Multiply by .
Step 1.5.5.3.2
Multiply by .
Step 1.5.5.3.3
Multiply by .
Step 1.5.5.3.4
Multiply by by adding the exponents.
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Step 1.5.5.3.4.1
Move .
Step 1.5.5.3.4.2
Multiply by .
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Step 1.5.5.3.4.2.1
Raise to the power of .
Step 1.5.5.3.4.2.2
Use the power rule to combine exponents.
Step 1.5.5.3.4.3
Add and .
Step 1.5.5.3.5
Rewrite using the commutative property of multiplication.
Step 1.5.5.3.6
Multiply by by adding the exponents.
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Step 1.5.5.3.6.1
Move .
Step 1.5.5.3.6.2
Multiply by .
Step 1.5.5.3.7
Multiply by .
Step 1.5.5.3.8
Multiply by .
Step 1.5.5.4
Add and .
Step 1.5.5.5
Subtract from .
Step 1.5.5.6
Move .
Step 1.5.5.7
Move .
Step 1.5.5.8
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 the left side of the equation.
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Step 1.7.1.1
Regroup terms.
Step 1.7.1.2
Factor out of .
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Step 1.7.1.2.1
Factor out of .
Step 1.7.1.2.2
Rewrite as .
Step 1.7.1.2.3
Factor out of .
Step 1.7.1.3
Rewrite as .
Step 1.7.1.4
Since both terms are perfect cubes, factor using the difference of cubes formula, where and .
Step 1.7.1.5
Factor.
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Step 1.7.1.5.1
Simplify.
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Step 1.7.1.5.1.1
Move to the left of .
Step 1.7.1.5.1.2
Raise to the power of .
Step 1.7.1.5.2
Remove unnecessary parentheses.
Step 1.7.1.6
Factor out of .
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Step 1.7.1.6.1
Factor out of .
Step 1.7.1.6.2
Factor out of .
Step 1.7.1.6.3
Factor out of .
Step 1.7.1.7
Factor out of .
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Step 1.7.1.7.1
Factor out of .
Step 1.7.1.7.2
Factor out of .
Step 1.7.1.7.3
Factor out of .
Step 1.7.1.8
Apply the distributive property.
Step 1.7.1.9
Simplify.
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Step 1.7.1.9.1
Rewrite as .
Step 1.7.1.9.2
Multiply by .
Step 1.7.1.9.3
Multiply by .
Step 1.7.1.10
Add and .
Step 1.7.1.11
Factor.
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Step 1.7.1.11.1
Factor by grouping.
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Step 1.7.1.11.1.1
For a polynomial of the form , rewrite the middle term as a sum of two terms whose product is and whose sum is .
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Step 1.7.1.11.1.1.1
Factor out of .
Step 1.7.1.11.1.1.2
Rewrite as plus
Step 1.7.1.11.1.1.3
Apply the distributive property.
Step 1.7.1.11.1.1.4
Multiply by .
Step 1.7.1.11.1.2
Factor out the greatest common factor from each group.
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Step 1.7.1.11.1.2.1
Group the first two terms and the last two terms.
Step 1.7.1.11.1.2.2
Factor out the greatest common factor (GCF) from each group.
Step 1.7.1.11.1.3
Factor the polynomial by factoring out the greatest common factor, .
Step 1.7.1.11.2
Remove unnecessary parentheses.
Step 1.7.2
If any individual factor on the left side of the equation is equal to , the entire expression will be equal to .
Step 1.7.3
Set equal to and solve for .
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Step 1.7.3.1
Set equal to .
Step 1.7.3.2
Add to both sides of the equation.
Step 1.7.4
Set equal to and solve for .
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Step 1.7.4.1
Set equal to .
Step 1.7.4.2
Solve for .
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Step 1.7.4.2.1
Subtract from both sides of the equation.
Step 1.7.4.2.2
Divide each term in by and simplify.
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Step 1.7.4.2.2.1
Divide each term in by .
Step 1.7.4.2.2.2
Simplify the left side.
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Step 1.7.4.2.2.2.1
Dividing two negative values results in a positive value.
Step 1.7.4.2.2.2.2
Divide by .
Step 1.7.4.2.2.3
Simplify the right side.
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Step 1.7.4.2.2.3.1
Divide by .
Step 1.7.5
Set equal to and solve for .
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Step 1.7.5.1
Set equal to .
Step 1.7.5.2
Add to both sides of the equation.
Step 1.7.6
The final solution is all the values that make true.
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 3.2.1.2.5
Multiply by .
Step 3.2.1.2.6
Multiply by .
Step 3.2.1.2.7
Multiply by .
Step 3.2.1.2.8
Multiply by .
Step 3.2.1.2.9
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
Add and .
Step 3.2.3.5
Subtract from .
Step 3.2.3.6
Add and .
Step 3.2.3.7
Add and .
Step 3.2.3.8
Add and .
Step 3.2.3.9
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
Multiply each element of by to make the entry at a .
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Step 3.3.2.1.1
Multiply each element of by to make the entry at a .
Step 3.3.2.1.2
Simplify .
Step 3.3.2.2
Perform the row operation to make the entry at a .
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Step 3.3.2.2.1
Perform the row operation to make the entry at a .
Step 3.3.2.2.2
Simplify .
Step 3.3.2.3
Perform the row operation to make the entry at a .
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Step 3.3.2.3.1
Perform the row operation to make the entry at a .
Step 3.3.2.3.2
Simplify .
Step 3.3.2.4
Multiply each element of by to make the entry at a .
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Step 3.3.2.4.1
Multiply each element of by to make the entry at a .
Step 3.3.2.4.2
Simplify .
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
Find the eigenvector using the eigenvalue .
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Step 4.1
Substitute the known values into the formula.
Step 4.2
Simplify.
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Step 4.2.1
Subtract the corresponding elements.
Step 4.2.2
Simplify each element.
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Step 4.2.2.1
Subtract from .
Step 4.2.2.2
Subtract from .
Step 4.2.2.3
Subtract from .
Step 4.2.2.4
Subtract from .
Step 4.2.2.5
Subtract from .
Step 4.2.2.6
Subtract from .
Step 4.2.2.7
Subtract from .
Step 4.2.2.8
Subtract from .
Step 4.2.2.9
Subtract from .
Step 4.3
Find the null space when .
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Step 4.3.1
Write as an augmented matrix for .
Step 4.3.2
Find the reduced row echelon form.
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Step 4.3.2.1
Multiply each element of by to make the entry at a .
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Step 4.3.2.1.1
Multiply each element of by to make the entry at a .
Step 4.3.2.1.2
Simplify .
Step 4.3.2.2
Perform the row operation to make the entry at a .
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Step 4.3.2.2.1
Perform the row operation to make the entry at a .
Step 4.3.2.2.2
Simplify .
Step 4.3.2.3
Perform the row operation to make the entry at a .
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Step 4.3.2.3.1
Perform the row operation to make the entry at a .
Step 4.3.2.3.2
Simplify .
Step 4.3.2.4
Multiply each element of by to make the entry at a .
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Step 4.3.2.4.1
Multiply each element of by to make the entry at a .
Step 4.3.2.4.2
Simplify .
Step 4.3.2.5
Perform the row operation to make the entry at a .
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Step 4.3.2.5.1
Perform the row operation to make the entry at a .
Step 4.3.2.5.2
Simplify .
Step 4.3.2.6
Perform the row operation to make the entry at a .
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Step 4.3.2.6.1
Perform the row operation to make the entry at a .
Step 4.3.2.6.2
Simplify .
Step 4.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 4.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 4.3.5
Write the solution as a linear combination of vectors.
Step 4.3.6
Write as a solution set.
Step 4.3.7
The solution is the set of vectors created from the free variables of the system.
Step 5
Find the eigenvector using the eigenvalue .
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Step 5.1
Substitute the known values into the formula.
Step 5.2
Simplify.
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Step 5.2.1
Simplify each term.
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Step 5.2.1.1
Multiply by each element of the matrix.
Step 5.2.1.2
Simplify each element in the matrix.
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Step 5.2.1.2.1
Multiply by .
Step 5.2.1.2.2
Multiply by .
Step 5.2.1.2.3
Multiply by .
Step 5.2.1.2.4
Multiply by .
Step 5.2.1.2.5
Multiply by .
Step 5.2.1.2.6
Multiply by .
Step 5.2.1.2.7
Multiply by .
Step 5.2.1.2.8
Multiply by .
Step 5.2.1.2.9
Multiply by .
Step 5.2.2
Add the corresponding elements.
Step 5.2.3
Simplify each element.
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Step 5.2.3.1
Subtract from .
Step 5.2.3.2
Add and .
Step 5.2.3.3
Add and .
Step 5.2.3.4
Add and .
Step 5.2.3.5
Subtract from .
Step 5.2.3.6
Add and .
Step 5.2.3.7
Add and .
Step 5.2.3.8
Add and .
Step 5.2.3.9
Subtract from .
Step 5.3
Find the null space when .
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Step 5.3.1
Write as an augmented matrix for .
Step 5.3.2
Find the reduced row echelon form.
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Step 5.3.2.1
Perform the row operation to make the entry at a .
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Step 5.3.2.1.1
Perform the row operation to make the entry at a .
Step 5.3.2.1.2
Simplify .
Step 5.3.2.2
Perform the row operation to make the entry at a .
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Step 5.3.2.2.1
Perform the row operation to make the entry at a .
Step 5.3.2.2.2
Simplify .
Step 5.3.2.3
Multiply each element of by to make the entry at a .
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Step 5.3.2.3.1
Multiply each element of by to make the entry at a .
Step 5.3.2.3.2
Simplify .
Step 5.3.3
Use the result matrix to declare the final solution to the system of equations.
Step 5.3.4
Write a solution vector by solving in terms of the free variables in each row.
Step 5.3.5
Write the solution as a linear combination of vectors.
Step 5.3.6
Write as a solution set.
Step 5.3.7
The solution is the set of vectors created from the free variables of the system.
Step 6
The eigenspace of is the list of the vector space for each eigenvalue.