Does the eigenvalue of the orthogonal matrix have to be 1 or 1?

Be. ,= (aα,aα) = (aα)^t(aα) = α^ta^taα

tα = (α

So there are 2( ,= (

And because ≠0, so ( 0.

So 2 = 1

So = 1

i.e. orthogonal matrices.

Eigenvalues can only be 1 or -1.

If AAT=E (E is the identity matrix.

AT stands for "transpose matrix of matrix A") or ATA=E, then the nth order real matrix A is called an orthogonal matrix. An orthogonal matrix is a unitary matrix that is specialized in real numbers.

Therefore it always belongs to the regular matrix. Although we are only considering real matrices here, this definition can be used for matrices whose elements come from any domain.

After all, the orthogonal matrix is an inner product.

Naturally, 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 this orthogonal matrix are real numbers) can be seen as a special kind of unitary matrix, but there is also a complex orthogonal matrix, which is not a unitary matrix.

Not necessarily, the matrix with (0 -1) (1 0) as the row vector is an orthogonal matrix, and the eigenvalues are i, -i

a inverses a e and then adds the determinant directly to both sides.

Here's why:Let be an orthogonal matrix.

The eigenvalue of a, x is the eigenvector of a, which belongs to the eigenvalue.

That is, there is ax = x, and x ≠ 0.

Transpose both sides to get x ta t = x t.

So x ta tax = 2x tx.

Because a is an orthogonal matrix, a ta=e.

So x tx = 2x tx.

From x≠0 we know that x tx is a non-zero number. Demolition.

Hence 2=1.

So 1 or 1.

1. In matrix theory, the orthogonal matrix of real numbers is the square matrix q, and its transpose matrix is its inverse matrix.

If the determinant of the orthogonal matrix.

is 1, and the fierce draft is called a special orthogonal matrix.

2. Sufficient and necessary conditions for the orthogonal of phalanx a.

A row (column) vector group is a unit orthogonal vector group.

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

4. The sufficient and necessary condition for a to be an orthogonal matrix is: the row vector group of a is orthogonal in pairs and is a unit vector.

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

6. The orthogonal phalanx is Euclidean space.

Transition matrix from the middle standard orthogonal basis to the standard orthogonal basis.

The eigenvalue of the orthogonal matrix must be 1 or -1.

αaα,aα) aα)^t(aα) ta^taα

t = so there is 2( and because ≠0, so ( 0

So 2 = 1

So = 1

That is, the eigenvalues of orthogonal matrices can only be 1 or -1.

The characteristics of orthogonal matrices are as follows:

1. The real square matrix is orthogonal, and if and only if its columns form the orthogonal canonical basis of Euclidean space r with an ordinary Euclidean dot product, it is true and only if its rows form the orthogonal basis of r.

2. The determinant of any orthogonal matrix is +1 or 1. This can be derived from the following basic facts about the determinant: (Note: The reverse is not true; There is a +1 determinant that does not guarantee orthogonality, even with orthogonal columns, which can be confirmed by the following counterexample. ）

3. For the permutation matrix, whether the determinant is +1 or 1 matches the sign of whether the permutation is even or odd, and the determinant is the alternating function of the rows.

4. What is stronger than the determinant constraint is that orthogonal matrices can always be a complete set of eigenvalues that can be diagonalized on complex numbers to show eigenvalues, and they must all have (complex) absolute values of 1.

Must be equal to 1 or -1.

The proof is as follows: let be the eigenvalue of the orthogonal matrix a, and x is the eigenvector of a eigenvalue of a, i.e., there is ax = x, and x≠0. Take the transposition on both sides, we get x ta t = x t so x ta tax = 2x tx, because a is an orthogonal matrix, so a ta=e, so x tx = 2x tx, from x≠0 we know that x tx is a non-zero number, so 2=1, so =1 or -1.

If aat=e (e is the identity matrix, at represents the "transpose matrix of matrix a". or ata=e, then the nth order real matrix a is called an orthogonal matrix, and if a is an orthogonal matrix, the following conditions are satisfied:

1. The rows of at are unit vectors and orthogonal in pairs.

2. The columns of at are unit vectors and quadrature in pairs.

3、(ax,ay)=(x,y)x,y∈r。

4、|a|=1 or -1.

5. Orthogonal matrices are usually represented by the letter q.

The role of orthogonal matrices.

Numerical analysis naturally takes advantage of the many numerical linear algebraic properties of orthogonal matrices. For example, it is often necessary to compute the orthogonal basis of the space, or the orthogonal change of the basis; Both take the form of orthogonal matrices. Having a determinant of 1 and all eigenvalues of modulus 1 is very advantageous for numerical stability.

An implication is that the conditional number is 1 (which is extremely small), so the error is not amplified when multiplying the orthogonal matrix. Many algorithms use orthogonal matrices for this purpose, such as householder reflection and givens rotation. It's helpful not only that the orthogonal matrix is reversible, but also that its inverse matrix is inherently cost-free, requiring only the index (subscript) to be swapped.

Permutation is fundamental to the success of many algorithms, including the computationally heavy Gaussian elimination method with partialpivoting (where permutation is used to determine the fulcrum). But they rarely appear explicitly as matrices; Their special form allows for more limited representations, such as a list of n indexes.

Proof: Let be the eigenvalue of the orthogonal matrix a, which is the eigenvector of the eigenvalue of a, that is, there is (a co-choke).'a =e,aα=λα,0.

On both sides of the equation a = take the co-choke transposition ( co-choke ).'(a)'= ( co-choke) ( co-choke).'.

Multiply a on both sides of the equation to get :

Co-choke).'(a)'a = (co-choke) (co-choke).'a i.e. (co-choke).'Co-choke).'(a)'a = (co-choke) (co-choke).'α

So [(co-choke) 1 ] co-choke).'α = 0.

Because ≠0

So ( co-choke) = 1

i.e. the modulo of is 1

I forgot the definition of orthogonal matrices on complex number fields, and I think it should be (a common choke).'a = e.

Must be equal to 1 or -1. If AAT=E (E is the identity matrix, AT stands for "Transpose Matrix of Matrix A") or ATA=E, then the nth order real matrix A is called an orthogonal matrix.

Introduction

A refiection, also known as a mirror reflection or mirror transformation, is similar to the shadow of an object in a mirror. Given a straight line on a two-dimensional plane, we can make a mirror reflection of the straight line.

Rotation inversion (rotoinversion): axis (0, -3 5, 4 5), angle 90°; Displacement axes, etc.