Let the eigenvalue of the third-order real symmetric matrix A be -1,1,1. Let the eigenvector X=(-1,1,1) corresponding to the eigenvalue -1, find A

Let the eigenvalue of the third-order real symmetric matrix A be -1,1,1. Let the eigenvector X=(-1,1,1) corresponding to the eigenvalue -1, find A

Orthogonally known from eigenvectors of real symmetric matrices belonging to different eigenvalues
The eigenvector a1=(-1,1,1)' corresponding to the eigenvalue-1 is orthogonal to the eigenvector belonging to the eigenvalue 1 and X=(x1, x2, x3)'
I.e.-x1+x2+x3=0.
Solve a basic solution system a2=(1,0,1)', a3=(1,1,0)'.
Orthogonalization of a2, a3 gives b1=(1,0,1)', b2=(1/2,1,-1/2)'=(1/2)(1,2-1)'.
Unit a1, b2, b3
C1=(-1/√3,1/√3,1/√3)', c2=(1/√2,0,1/√2)', c3=(1/√6,2/√6,-1/√6)'.
Let P=(c1, c2, c3)=
-1/√3 1/√2 1/√6
1/√3 0 2/√6
1/√3 1/√2 -1/√6
Then P is the orthogonal matrix, satisfying P^-1AP=diag (-1,1,1)
So A=Pdiag (-1,1,1) P^-1=
1/3 2/3 2/3
2/3 1/3 -2/3
2/3 -2/3 1/3
=(1/3)*[ Raise 1/3 to look better]
1 2 2
2 1 -2
2 -2 1

Let the eigenvalue of the third-order real symmetric matrix A be -1,1,1. Let the eigenvector X=(0,1,1) corresponding to the eigenvalue -1. I would like to ask the following basic solution: a2=(1,0,0)^T, a3=(0,1,-1)^T. Let the eigenvalue of the third-order real symmetric matrix A be -1,1,1. Let the eigenvector X=(0,1,1) corresponding to the eigenvalue -1. I want to ask the following basic solution: a2=(1,0,0)^T, a3=(0,1,-1)^T.

The system of equations is x2+x3=0
X1, x2 are regarded as free unknown quantities, and the basic solution a2=(1,0,0)^T, a3=(0,1,-1)^T is obtained by taking 1,0 and 0,1, respectively.
(1,1,1)^T is the solution
(0,0,0)^T No
The basic solution must be linearly independent

Let the eigenvalue of the 3rd order real symmetric matrix A be the transposition of the eigenvector corresponding to -1,1,1,-1 be (0,1,1). Let the eigenvector belonging to the eigenvalue 1 be (x1, x2, x3)^T Because real symmetric matrices belong to orthogonal eigenvectors with different eigenvalues Therefore,(x1, x2, x3)^T is orthogonal to a1=(0,1,1)^T. I.e. x2+x3=0. The basic solution is: a2=(1,0,0)^T, a3=(0,1,-1)^T I want to know how to find the basic solution system a2, a3, x2+x3=0. Corresponding matrix (0 1 1), The free unknown quantity should be x2

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Let λ1=6λ2=λ3=3,ξ1=(1,1,1) be a eigenvector of eigenvalue 6 for A Let λ1=6λ2=λ3=3,ξ1=(1,1,1) be the transposition of the eigenvector of the eigenvalue 6 of the symmetric matrix A

In the symmetric matrix, the eigenvectors corresponding to different eigenvalues are orthogonal.
Therefore, the space formed by the corresponding eigenvectors is the solution space of (1,1,1) x=0.
Let one of its basic solutions be (-1,1,0)^T,(-1,0,1)^T
Joint (1,1,1)^T, orthogonalize three vector gauge, and obtain vector p1, p2, p3, denoted P=(p1, p2, p3), denoted B=
6 0 0
0 3 0
0 0 3
Then P^TAP=B
So A=PBP^T

Arrowhead Today, I had to learn acceleration. The teacher talked about horizontal throwing, circular motion, and vector operation. I don't know what the arrow is. The teacher drew a circle in the coordinate system, saying acceleration direction, centripetal force,45 degrees, and the same force. He spoke with special passion, and I listened with special helplessness. Please help me explain a few words about the teacher, arrow understanding. I understand the concept. It's better to have a picture. Arrowhead Today, I had to learn acceleration. The teacher talked about horizontal throwing, circular motion and vector operation. I don't quite understand what the arrow is. The teacher drew a circle in the coordinate system, saying what acceleration direction, centripetal force,45 degrees, the same force. He talked with special passion, and I listened with special helplessness. Please help me explain a few words about the teacher, arrow understanding. I understand the concept. It's better to have a picture. It's clear.

The circle drawn by your teacher represents the trajectory of an object's motion (circular motion). The speed of an object is the same at every point on the circumference (just the same size, the speed direction is different), but because of different directions, it is still acceleration. The acceleration direction is inward along the radius direction, and the size is "...

High School Physics Compulsory One Vector Operation Find the vector and vector operation detailed explanation. Also want to ask whether the vector operation is important at this stage, whether you must master parallelogram algorithm, etc. Understanding ability is not too good, hope to explain a little easier to understand.

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