Linear Algebra Examples

Find the Eigenvectors/Eigenspace [[-2,-1,-1],[-2,-3,-2],[2,2,1]]
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 row 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
Evaluate .
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Step 1.5.2.1
The determinant of a matrix can be found using the formula .
Step 1.5.2.2
Simplify the determinant.
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Step 1.5.2.2.1
Simplify each term.
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Step 1.5.2.2.1.1
Expand using the FOIL Method.
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Step 1.5.2.2.1.1.1
Apply the distributive property.
Step 1.5.2.2.1.1.2
Apply the distributive property.
Step 1.5.2.2.1.1.3
Apply the distributive property.
Step 1.5.2.2.1.2
Simplify and combine like terms.
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Step 1.5.2.2.1.2.1
Simplify each term.
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Step 1.5.2.2.1.2.1.1
Multiply by .
Step 1.5.2.2.1.2.1.2
Multiply by .
Step 1.5.2.2.1.2.1.3
Multiply by .
Step 1.5.2.2.1.2.1.4
Rewrite using the commutative property of multiplication.
Step 1.5.2.2.1.2.1.5
Multiply by by adding the exponents.
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Step 1.5.2.2.1.2.1.5.1
Move .
Step 1.5.2.2.1.2.1.5.2
Multiply by .
Step 1.5.2.2.1.2.1.6
Multiply by .
Step 1.5.2.2.1.2.1.7
Multiply by .
Step 1.5.2.2.1.2.2
Subtract from .
Step 1.5.2.2.1.3
Multiply by .
Step 1.5.2.2.2
Add and .
Step 1.5.2.2.3
Reorder and .
Step 1.5.3
Evaluate .
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Step 1.5.3.1
The determinant of a matrix can be found using the formula .
Step 1.5.3.2
Simplify the determinant.
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Step 1.5.3.2.1
Simplify each term.
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Step 1.5.3.2.1.1
Apply the distributive property.
Step 1.5.3.2.1.2
Multiply by .
Step 1.5.3.2.1.3
Multiply by .
Step 1.5.3.2.1.4
Multiply by .
Step 1.5.3.2.2
Add and .
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
Multiply by .
Step 1.5.4.2.1.2
Apply the distributive property.
Step 1.5.4.2.1.3
Multiply by .
Step 1.5.4.2.1.4
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
Subtract from .
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 by adding the exponents.
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Step 1.5.5.3.3.1
Move .
Step 1.5.5.3.3.2
Multiply by .
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Step 1.5.5.3.3.2.1
Raise to the power of .
Step 1.5.5.3.3.2.2
Use the power rule to combine exponents.
Step 1.5.5.3.3.3
Add and .
Step 1.5.5.3.4
Rewrite using the commutative property of multiplication.
Step 1.5.5.3.5
Multiply by by adding the exponents.
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Step 1.5.5.3.5.1
Move .
Step 1.5.5.3.5.2
Multiply by .
Step 1.5.5.3.6
Multiply by .
Step 1.5.5.3.7
Multiply by .
Step 1.5.5.4
Subtract from .
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
Factor out of .
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Step 1.7.1.1.1
Factor out of .
Step 1.7.1.1.2
Factor out of .
Step 1.7.1.1.3
Factor out of .
Step 1.7.1.1.4
Rewrite as .
Step 1.7.1.1.5
Factor out of .
Step 1.7.1.1.6
Factor out of .
Step 1.7.1.1.7
Factor out of .
Step 1.7.1.2
Factor using the rational roots test.
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Step 1.7.1.2.1
If a polynomial function has integer coefficients, then every rational zero will have the form where is a factor of the constant and is a factor of the leading coefficient.
Step 1.7.1.2.2
Find every combination of . These are the possible roots of the polynomial function.
Step 1.7.1.2.3
Substitute and simplify the expression. In this case, the expression is equal to so is a root of the polynomial.
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Step 1.7.1.2.3.1
Substitute into the polynomial.
Step 1.7.1.2.3.2
Raise to the power of .
Step 1.7.1.2.3.3
Raise to the power of .
Step 1.7.1.2.3.4
Multiply by .
Step 1.7.1.2.3.5
Add and .
Step 1.7.1.2.3.6
Multiply by .
Step 1.7.1.2.3.7
Subtract from .
Step 1.7.1.2.3.8
Add and .
Step 1.7.1.2.4
Since is a known root, divide the polynomial by to find the quotient polynomial. This polynomial can then be used to find the remaining roots.
Step 1.7.1.2.5
Divide by .
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Step 1.7.1.2.5.1
Set up the polynomials to be divided. If there is not a term for every exponent, insert one with a value of .
++++
Step 1.7.1.2.5.2
Divide the highest order term in the dividend by the highest order term in divisor .
++++
Step 1.7.1.2.5.3
Multiply the new quotient term by the divisor.
++++
++
Step 1.7.1.2.5.4
The expression needs to be subtracted from the dividend, so change all the signs in
++++
--
Step 1.7.1.2.5.5
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
++++
--
+
Step 1.7.1.2.5.6
Pull the next terms from the original dividend down into the current dividend.
++++
--
++
Step 1.7.1.2.5.7
Divide the highest order term in the dividend by the highest order term in divisor .
+
++++
--
++
Step 1.7.1.2.5.8
Multiply the new quotient term by the divisor.
+
++++
--
++
++
Step 1.7.1.2.5.9
The expression needs to be subtracted from the dividend, so change all the signs in
+
++++
--
++
--
Step 1.7.1.2.5.10
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
+
++++
--
++
--
+
Step 1.7.1.2.5.11
Pull the next terms from the original dividend down into the current dividend.
+
++++
--
++
--
++
Step 1.7.1.2.5.12
Divide the highest order term in the dividend by the highest order term in divisor .
++
++++
--
++
--
++
Step 1.7.1.2.5.13
Multiply the new quotient term by the divisor.
++
++++
--
++
--
++
++
Step 1.7.1.2.5.14
The expression needs to be subtracted from the dividend, so change all the signs in
++
++++
--
++
--
++
--
Step 1.7.1.2.5.15
After changing the signs, add the last dividend from the multiplied polynomial to find the new dividend.
++
++++
--
++
--
++
--
Step 1.7.1.2.5.16
Since the remander is , the final answer is the quotient.
Step 1.7.1.2.6
Write as a set of factors.
Step 1.7.1.3
Factor using the AC method.
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Step 1.7.1.3.1
Factor using the AC method.
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Step 1.7.1.3.1.1
Consider the form . Find a pair of integers whose product is and whose sum is . In this case, whose product is and whose sum is .
Step 1.7.1.3.1.2
Write the factored form using these integers.
Step 1.7.1.3.2
Remove unnecessary parentheses.
Step 1.7.1.4
Factor.
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Step 1.7.1.4.1
Combine like factors.
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Step 1.7.1.4.1.1
Raise to the power of .
Step 1.7.1.4.1.2
Raise to the power of .
Step 1.7.1.4.1.3
Use the power rule to combine exponents.
Step 1.7.1.4.1.4
Add and .
Step 1.7.1.4.2
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
Solve for .
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Step 1.7.3.2.1
Set the equal to .
Step 1.7.3.2.2
Subtract from 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
Subtract from both sides of the equation.
Step 1.7.5
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
Add the corresponding elements.
Step 3.2.2
Simplify each element.
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Step 3.2.2.1
Add and .
Step 3.2.2.2
Add and .
Step 3.2.2.3
Add and .
Step 3.2.2.4
Add and .
Step 3.2.2.5
Add and .
Step 3.2.2.6
Add and .
Step 3.2.2.7
Add and .
Step 3.2.2.8
Add and .
Step 3.2.2.9
Add and .
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 3.3.3
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
Simplify each term.
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Step 4.2.1.1
Multiply by each element of the matrix.
Step 4.2.1.2
Simplify each element in the matrix.
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Step 4.2.1.2.1
Multiply by .
Step 4.2.1.2.2
Multiply by .
Step 4.2.1.2.3
Multiply by .
Step 4.2.1.2.4
Multiply by .
Step 4.2.1.2.5
Multiply by .
Step 4.2.1.2.6
Multiply by .
Step 4.2.1.2.7
Multiply by .
Step 4.2.1.2.8
Multiply by .
Step 4.2.1.2.9
Multiply by .
Step 4.2.2
Add the corresponding elements.
Step 4.2.3
Simplify each element.
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Step 4.2.3.1
Add and .
Step 4.2.3.2
Add and .
Step 4.2.3.3
Add and .
Step 4.2.3.4
Add and .
Step 4.2.3.5
Add and .
Step 4.2.3.6
Add and .
Step 4.2.3.7
Add and .
Step 4.2.3.8
Add and .
Step 4.2.3.9
Add and .
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
Swap with to put a nonzero entry at .
Step 4.3.2.2
Multiply each element of by to make the entry at a .
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Step 4.3.2.2.1
Multiply each element of by 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
The eigenspace of is the list of the vector space for each eigenvalue.