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To find the eigenvalues and eigenvectors of a third order matrix: to find the eigenvalues and eigenvectors of (1 2 3 2 1 3 3 3 6)
Let the eigenvalue of the matrix a be λ
be
|A-λE|=
1-λ 2 3
2 1-λ 3
3 6 - λ column 2 minus column 1
=
1-λ λ+1 3
2 -1-λ 3
3 06 - λ line 1 plus line 2
=
3-λ 0 6
2 -1-λ 3
30 6 - λ is expanded according to the second column
=(-1-λ)(λ²-9λ)=0
The solution is λ = 9,0 or - 1
When λ = 9,
A-9E=
-8 2 3
2 -8 3
3 3 - 3 line 1 plus line 2 × 4, line 3 divided by 3,
～
0 -30 15
2 -8 3
1 1 - 1 line 1 divided by - 15, line 2 minus line 3 multiplied by 2
～
0 2 -1
0 -10 5
1 1 - 1 line 2 plus line 1 × 5, line 1 multiplied by 1 / 2, line 3 minus line 1, exchange lines
～
1 0 -1/2
0 1 -1/2
0 0 0
Get the eigenvector (1,1,2) ^ t
When λ = 0,
A=
1 2 3
2 1 3
3 3 6 line 2 minus line 1 multiplied by 2, line 3 minus line 1 multiplied by 3
～
1 2 3
0 -3 -3
0 - 3 - 3 line 3 minus line 2, line 2 divided by - 3, line 1 minus line 2 multiplied by 2
～
1 0 1
0 1 1
0 0 0
Get the eigenvector (1,1, - 1) ^ t
When λ = - 1,
A+E=
2 2 3
2 2 3
3 3 7 line 2 minus line 1, line 3 minus line 1 × 3 / 2
～
2 2 3
0 0 0
Divide line 3 by 2.5, subtract line 3 × 3 from line 1, and exchange lines 2 and 3
～
2 2 0
0 0 1
0 0 0
Get the eigenvector (1, - 1,0) ^ t
So the eigenvalues of this matrix are 9,0, - 1
The corresponding eigenvectors are: (1,1,2) ^ t, (1,1, - 1) ^ t, (1, - 1,0) ^ t
Linear Algebra: a matrix of order n has the same eigenvalues as its transpose matrix a '
It is proved that | λ I-A | = | λ I-A '|
Therefore, the eigenvalues of matrix A and transpose matrix of matrix A are the same
Question: in the eigenvalue (λ I-A) α = 0 (λ I-A) is a determinant, not | λ I-A | is a numerical value, how can it be the same
Because the eigenvalue is the root of the characteristic equation | λ I-A | = 0, it is necessary to prove that the eigenvalues are the same as long as the characteristic equations are the same
Let B = λ I-A, DETB = detb'according to determinant knowledge
That is, | λ I-A | = | (λ I-A) '| = | λ I-A' |, so the eigenvalues of a and a 'are the same
Because D (arcsinx)=____ So arcsinx is____ A primitive function of
Because D (arcsinx)=__ dx/√(1-x²)__ So arcsinx is__ 1/√(1-x²)__ A primitive function of
Xiao Ming uses four identical rectangles and a small square to make a big square. The area of the big and small squares is 100 square centimeters and 4 square centimeters respectively
Do you know the length and width of the rectangle he used?
4 and 6
Given that a, B and C are three sides of △ ABC, if the equation AX ^ 2 + 2 √ (b ^ 2 + C ^ 2) * x + 2 (B + C) = 2A has two equal real roots, try to judge the shape of △ ABC
The equation AX ^ 2 + 2 √ (b ^ 2 + C ^ 2) * x + 2 (B + C-A) = 0 has two equal real roots
Then [2 √ (b ^ 2 + C ^ 2)] ^ 2-8a (B + C-A) = 0
That is, B ^ 2 + C ^ 2-2ab-2ac + 2A ^ 2 = 0
(a-b)^2+(c-a)^2=0
The above formula holds only when they are respectively zero
Then a = b = C
ABC is an equilateral triangle
The derivation of parametric equation x = acost y = bsint why DX -- = - asint dy
Why is there a x = acost x0cy = bsint x0c so x0cdx = - asint DT x0cdy = bCost DT
Derivation of X from t
dx=-asint dt
Derivation of Y from t
dy=bcost dt
2
dx/dy=-asint dt/bcost dt=-a/btant
dx=-a/btant dy
It can't be wrong
The derivative of the whole is equal to the constant times the derivative of the expression
How can five small squares make a big square
Divide the five small squares into three parts to form a square
My space has detailed answers and graphics to this question
cross:
Another similar problem:
There are eight same cylindrical square columns in the hall. The perimeter of the bottom is 2.4 meters and the height is 6 meters. Now we need to repaint them,
How much will it cost to paint all the square pillars
The total area of the column: 2.4 * 6 * 8 = 115.2 square meters. The price of different coatings is different. The labor cost is different in different areas. Consult the construction team or the paint seller
Add the appropriate operation symbol +. -. X.%, or parentheses between the following five 5S to make the following formula true
5 5 5 5 5=10
5x5-5-5-5=10
Is the existence of two partial derivatives x (x0, Y0), y (x0, Y0) of binary function f (x, y) at the point (x0, Y0) continuous?
What are the conditions
It is neither sufficient nor necessary
If f (x, y) = (XY) / (x + y) is not at the origin, let it be equal to zero at the origin
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