Regarding the question of orthogonal matrices, does the matrix of orthogonal transformation have to

Solution: det|λe-a|=|λ-1 2 0|=(λ-2)(λ5)(λ1)=0

The eigenvalues are 1=2, 2=5, 3=-1

For eigenvalues 1=2, solve the system of equations (2e-a) x=0 by.

2e-a=[1 2 0] [1 0 1 ]

One of its basic solutions is 1=(-2,1,2) t and unitifies 1 to obtain the unit eigenvector 1=(-2 3,1 3,2 3) t belonging to 1=2

For eigenvalues 2=5, solve the system of equations (5e-a) x=0 by.

5e-a=[4 2 0] [1 0 -1/2]

One of its basic solutions is 2=(1,-2,2) t and unitifies 2, and the unit eigenvector 1=(1 3,-2 3,2 3) t belonging to 2=5 is obtained

For eigenvalues 3=-1, solve the system of equations (-e-a) x=0 by.

e-a=[-2 2 0] [1 0 -2]

One of its basic solutions is 3=(2,2,1) t and unitifies 3 to obtain the unit eigenvector 3=(2 3,2 3,1 3) t belonging to 3=-1

So 1, 2, 3 are the orthogonalized unitized eigenvectors of a, the order matrix.

q=[ 1, 2, 3]=[-2 3 1 3 2 3], then q is the orthogonal matrix that is sought, and there is q -1aq=q taq=[2 ].

It's annoying, you know the steps, right?

The matrix of the orthogonal transformation must be an orthogonal matrix. Since the modulus length and angle of the vector are both defined by the inner product, the respective modulus length and their angle of the pair of vectors before and after the orthogonal transformation are unchanged. In particular, the standard orthogonal basis remains the standard orthogonal basis after orthogonal transformation.

In finite-dimensional space, the matrix of orthogonal transformations under the standard orthogonal basis is represented as an orthogonal matrix, and all its rows and all columns also form a set of standard orthogonal bases of v. Because the determinant of an orthogonal matrix can only be +1 or 1, the determinant of an orthogonal transformation is +1 or 1.

Orthogonal transformations with determinants of +1 and 1 are called Class 1 (corresponding to rotation transformations) and Class 2 (corresponding to blemish rotation transformations), respectively. It can be seen that orthogonal transformations in Euclidean space contain only rotation, reflection, and their combination (i.e., flawed rotation).

Matrix interorthogonal is two vectors orthogonal, two vectors orthogonal means that their inner product is equal to zero, and the inner product of two vectors is the sum of the products of their corresponding components.

The concept of geometric vectors is abstracted in algebra to obtain a more general concept of vectors. 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.

Orthogonal vector analysis first appeared 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 , it is denoted as

1. The sufficient and necessary condition for the quadrature a is that the row (column) vector group of a is the unit orthogonal vector group;

2. The sufficient and necessary condition for the orthogonal of the phalanx a is that the n row (column) vectors of a a are a set of standard orthogonal bases of the n-dimensional vector space;

3. The sufficient and necessary conditions for a to be an orthogonal matrix are: the row vector group of a is orthogonal in pairs and is a unit vector;

4. The column vector group of a is also an orthogonal unit vector group;

5. The orthogonal square matrix is the transition matrix from the standard orthogonal basis to the standard orthogonal basis in Euclidean space.

An orthogonal matrix is a matrix of squares, and both row and column vectors are orthogonal unit vectors.

The row vectors are all orthogonal unit vectors, and any two rows are orthogonal to the two rows of points and the result is 0, and because they are unit vectors, the result of any row of points multiplied by themselves is 1.

For a 3x3 orthogonal matrix, each row is a 3-dimensional vector, and the geometric meaning of two 3-dimensional vectors orthogonal is that the two vectors are perpendicular to each other.

So the three rows of the 3x3 orthogonal matrix can be understood as the three coordinate axes in a 3D coordinate system, and the following are the 3 3 orthogonal matrix m, x1, x2, x3, x-axis y1, y2, y3, y-axis z1, z2, z3, z-axis.

The three axes represented by the identity matrix are the x, y, and z axes in the Cartesian coordinate system

1,0,0, x-axis0,1,0, y-axis0,0,1, z-axis.

The geometric meaning of a vector multiplied by a 3x3 orthogonal matrix is to transform this vector from the current coordinate system to the coordinate system represented by this matrix, for example, the following matrix m1,0,1,0,1,0,0,0,0,0,1, a vector (1,2,3) is right multiplied by this matrix m1 to get a new vector (2,1,3), that is, the original vector is transformed from the original coordinate system to a new coordinate system.

The x-axis of the new coordinate system is (0,1,0) in the original coordinate system, that is, it falls on the y-axis of the original coordinate system, and the new coordinate system is to reverse the x and y axes of the original coordinate system, so the orthogonal matrix m1 acts on the vector (1,2,3) and then reverses the x and y components of the vector.

The definition of orthogonal matrices, "row vectors and column vectors are orthogonal unit vectors", brings another benefit: the transposition of orthogonal matrices is the inverse of orthogonal matrices, which is much simpler than finding the inverse of ordinary matrices.

Let's explain why the transpose of an orthogonal matrix is the inverse of an orthogonal matrix

Or the orthogonal matrix m:

x1，x2，x3，／／rowxy1，y2，y3，／／rowyz1，z2，z3，／／rowz

Each row is a unit length vector, so each row dot multiplied by itself results in 1.

Any two lines are orthogonal and the result is 0.

The transpose matrix MT of the matrix M is:

x1, y1, z1, x2, y2, z2, x3, y3, z3, two matrices multiplied mmul m mt:

rowx rowx, rowx rowy, rowx rowz, rowy rowx, rowy rowy, rowy rowz, rowz rowx, rowz rowy, rowz rowz, the point multiplies itself by 1, and the point multiplies by other rows results are 0, so mmul is equal to the identity matrix.

1,0,0,0,1,0,0,0,0,1, the definition of the inverse matrix is that the inverse matrix multiplied by the original matrix equals the identity matrix, so the transposition of the orthogonal matrix is the inverse of the orthogonal matrix.

The definition of an orthogonal matrix refers to a matrix whose transposition is equal to the inverse, and the properties are inverse and orthogonal matrix, and the product is also orthogonal matrix.

Further information is as follows:

If AAT=E (E is the identity matrix, and AT represents the "transposed matrix of matrix A") or ATA=E, then the nth-order real matrix A is called the orthogonal moment. Orthogonal matrices are unitary matrices that specialize in real numbers and therefore always belong to regular matrices. Although we are only considering real matrices here, this definition can be used for matrices whose elements come from any domain.

After all, orthogonal matrices are naturally derived from the inner product, so for matrices of complex numbers, this leads to a normalization requirement. Orthogonal matrices are not necessarily real matrices. A real orthogonal matrix (i.e., all elements in the orthogonal matrix are real numbers) can be regarded as a special kind of unitary matrix, but there is also a complex orthogonal matrix, which is not a unitary matrix.

The identity transformation is to turn an analytic formula into another analytic formula with its identity The use of identity transformation is often when the problem encountered is more complicated and difficult to start, the problem to be solved is simplified through the identity transformation, from the unknown to the known, and finally the problem is solved.

Regardless of the dimensions, it is always possible to classify orthogonal matrices as pure rotation or not, but for 3 3 matrices and higher dimensional matrices it is much more complicated than reflection. For example, it represents the inversion by the origin and the rotation inversion with respect to the z-axis (reflection in the x-y plane after 90° counterclockwise rotation, or inversion to the origin after 270° counterclockwise).

Rotation has also become more complex; They are no longer depicted by a single corner and may affect more than one planar subspace. Although the 3-3 rotation matrix is often described in terms of an axis and an angle, the existence of the axis of rotation in this dimension is contingent in nature and does not apply to other dimensions by hand. However, generally applicable basic building blocks such as displacement, reflection, and rotation can meet these conditions.

What is an orthogonal matrix.

The matrix that satisfies the formula is the orthogonal matrix, so what characteristics does the orthogonal matrix have?

Denote a matrix consisting of row vectors, then.

According to the formula, it can be known.

Hence is a unit vector.

Therefore orthogonal with .

Conclusion: The row (column) vectors in the orthogonal matrix beam are orthogonal unit vectors.