How to orthogonalize two vectors, the unit vector orthogonalization formula

Vector orthogonalization, symmetry matrix diagonalization depends on whether the problem requires an orthogonal matrix, quadratic standard type allows the orthogonal matrix to be found when the orthogonal transformation is made—if the eigenvalue is obtained.

If they are not equal, only the eigenvectors corresponding to them are needed.

Unitization (reason: real symmetric matrix.

Orthogonal of eigenvectors with different eigenvalues) Second, if the eigenvalues are equal, for example, a1=a2=a3=2, then the eigenvectors corresponding to eigenvalues equal to more than 2 should be orthogonal first, and then normalized (Schmidt orthogonalization.

For example, let b a2 + ta1For b a1, it must be (a2+ta1)·a1 0, i.e.: a2·a1+ta1·a1 0 t -(a2·a1) (a1·a1) -1) 2 1 2

b＝（0 2 1）t+（1/2(1 0 -1)t＝（1/2,2,1/2）t

a1,a2 a1,b [Vector group a1,a2 is equivalent to vector group a1,b, which is an orthogonal group].

This process is the orthogonalization of vector groups.

Gram Schmidt's principle of orthogonalization, see wiki for details.

In particular, for the two vectors a1 and a2, the orthogonal formula can be obtained, b1=a1, b2=a2-k b1

where k=(a2, b1) (b1, b1)

For bidirectional quantity orthogonalization, it is equivalent to the triangle principle (right angles).

Orthogonalization, unitization is the vectorization of this vector as a unit vector.

For example, the vector (1,2,3) is normalized as: [1 root number under (1 2+2 2+3 2), 2 root number under (1 2+2 2+3 2), 3 root number under (1 2+2 2+3 2)]=1 root number 14, 2 root number 14, 3 root number 14).

Eigenvectors of linear transformations.

A non-zero vector that does not change the direction under the transformation, or is simply multiplied by a scale factor. The eigenvalue corresponding to the eigenvector.

is the scale factor it multiplies.

A feature space is a space made up of all feature vectors with the same eigenvalues, including zero vectors, but it should be noted that the zero vector itself is not a eigenvector. The principal eigenvector of a linear transformation is the eigenvector corresponding to the maximum eigenvalue. The geometric order of the eigenvalues is the dimensionality of the corresponding eigenspace.

Extended data: Assuming that it is a linear transformation, then v can be represented by a set of bases of the vector space in which it is located: where vi is the projection (i.e., coordinates) of the vector on the basis vector, and here the vector space is assumed to be n-dimensional.

From this, it can be represented directly as a coordinate vector. With basis vectors, linear transformations can also be multiplied with a simple matrix.

Denote. Its eigenfunction satisfies the following eigenvalue equation: where is the eigenvalue corresponding to the function.

Such a function of time, if =0, it will not be dismantled, if it is positive, it will increase proportionally, and if it is negative, it will decay proportionally. For example, the idealized total number of rabbits reproduces faster where there are more rabbits, satisfying a positive eigenvalue equation.

If a is an n n matrix, then pa is a polynomial of the n order.

Therefore a has a maximum of n eigenvalues. In turn, the fundamental theorem of algebra says that the equation has exactly n roots, if the roots are double.

It is also counted. All odd-numbered polynomials must have a real root, so for odd n, each real matrix has at least one real eigenvalue. In the case of real matrices, for even numbers.

Vector orthogonalityThe formula is a=h l.

Orthogonalization refers to the process of transforming the linear independent Xiangou vector system into an orthogonal system. Let {xn} be the inner product space.

If there are finite or linearly independent vectors in h, then there must be a canonical orthogonal system in h such that for each positive integer.

n (when {xn} contains only m vectors, n m is required), xn is e1, e2 ,..., a linear combination of en.

Orthogonal property of two vectors: There are two n-dimensional vectors, and if their inner product is equal to zero, then these two vectors are said to be orthogonal to each other, which is denoted as obvious if

Note: Count the feasts.

For any group of vectors, it is either linearly independent or linearly correlated.

Contains zero vectors.

Any group of vectors is linearly related.

Groups of vectors containing vectors that are parallel to each other must be linearly correlated.

Vector groups are linearly correlated, so increasing the number of vectors does not change the correlation of vectors.

Local correlation, global correlation].

Vector groups are linearly independent, so reduce the number of vectors without changing the vector irrelevance. [The whole is irrelevant, the part is irrelevant].

The orthogonal calculation of two vectors is that their seepage inner product (dot product) is zero. Therefore, it is possible to tell if two vectors are orthogonal by calculating the dot product of them.

First, calculate the dot product of two vectors, i.e., multiply and add the numbers corresponding to their positions. Let the vectors a (a1,a2,a3) and vectors b (b1,b2,b3), then their dot products are: a·b a1b1 a2b2 a3b3.

Then determine whether the dot product of the two vectors is zero. If the dot product is zero, it means that the two shouting volumes are orthogonal; If the dot product is not zero, it means that the two vectors are not orthogonal. For example, the vector a (1,2,3) and the vector b (2,-1,0), then their dot product is:

a·b 1 2 2 (-1) 3 0 0, therefore, vectors a and b are orthogonal.

Considerations for computational vectors

1. The direction of the vector: The vector has a direction, and it is necessary to pay attention to the correctness of the direction, and the direction of the vector needs to be clear when calculating.

2. The size of the vector: The size of the vector is a representation of the length of the vector, and attention should be paid to calculating the correct vector size.

3. Addition and subtraction of vectors: The addition and subtraction of vectors need to meet the clear macro algebra operation rules of vectors, and the correctness of addition and subtraction should be paid attention to.

4. The quantity product of the vector: The quantity product of the vector is a kind of vector product, and it is necessary to pay attention to the calculation method and law of the quantity product.

5. Vector product of vectors: The vector product of vectors is a kind of vector product, and it is necessary to pay attention to the calculation method and law of vector product.

The product of the two vectors orthogonal is 0, so to determine whether the vector is orthogonal or not, look at whether the product of the two vectors is 0.

Do the inner product. That is, the corresponding components are multiplied and added up. If it is equal to 0, it is orthogonal, and the first one is 2*-2 + 1*1 +0*0 =-3, so it is not orthogonal, and the second 1*0+1*0 +0*1 =0 is orthogonal.

[ 1, 2]=a1b1+a2b2+a3b3+a4b4, that is, the inner product (point product) of two vectors, can be found by substituting the corresponding vectors, for example, when finding 2, substituting 1 and 2 into the above equation, the operation can be calculated.

Schmidt orthogonalization is a method for finding orthogonal bases in Euclidean space. Starting from the arbitrary linearly independent vector group 1, 2 and so on, m in Euclidean space, the orthogonal vector group 1, 2, m is obtained, so that 1, 2, m is equivalent to the vector group 1, 2, m, and then each vector in the orthogonal vector group is normalized to obtain a standard orthogonal vector group, which is called Schmidt orthogonalization.

Prove by mathematical induction:

The method of constructing a standard orthogonal vector group by using the vertical independent vector group of line speed file described above is the Schmidt orthogonalization method. An orthogonal vector group is a group of vectors consisting of a non-zero pair of orthogonal (i.e., with an inner product of 0). The concept of geometric vectors is abstracted in algebra to obtain a more stupid and general vector concept.

Vectors are defined as elements of a vector space, and it is important to note that these abstract vectors are not necessarily represented by pairs, nor do the concepts of size and direction apply. In a three-dimensional vector space, if the inner product of two vectors is zero, then the two vectors are said to be orthogonal. Orthogonality first appeared in vector analysis in three-dimensional space.

In other words, two vectors are orthogonal meaning that they are perpendicular to each other. If the vector is orthogonal to the mu size, it is denoted as

Vector orthogonalization generally uses Schmitt's method of orthogonalization.

After passing such calculations.

1，β2，……s is the orthogonal vector group.

is two vectors orthogonal representing the product of the two vectors as 0.

If there are two or more vectors, and their dot product is 0, then they are called orthogonal vectors to each other. In a two- or three-dimensional Euclidean space, two or three vectors are orthogonal to each other when they are at an angle of 90°. The set of orthogonal vectors is called an orthogonal vector group.

A1 and A2 are perpendicular to each other; The vertical product is 0

The orthogonal vector inner product is 0; So multiplying to 0 is orthogonal; So the first group is not orthogonal and the second group is orthogonal.

Two vectors orthogonal means that the product of these two vector vectors is equal to 0

Simply put, two vectors are perpendicular.