# Category: Trace of orthogonal matrix

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The trace of a matrix is the sum of its complex eigenvaluesand it is invariant with respect to a change of basis. This characterization can be used to define the trace of a linear operator in general. The trace is related to the derivative of the determinant see Jacobi's formula.

The trace is a linear mapping. That is, [2] [3]. A matrix and its transpose have the same trace: [2] [3] [4] : This follows immediately from the fact that transposing a square matrix does not affect elements along the main diagonal.

The trace of a square matrix which is the product of two matrices can be rewritten as the sum of entry-wise products of their elements. This means that the trace of a product of equal-sized matrices functions in a similar way to a dot product of vectors imagine A and B as long vectors with columns stacked on each other.

For this reason, generalizations of vector operations to matrices e. For real matrices A and Bthe trace of a product can also be written in the following forms:. More generally, the trace is invariant under cyclic permutationsthat is.

However, if products of three symmetric matrices are considered, any permutation is allowed, since:. Note that this is not true in general for more than three factors.

Unlike the determinantthe trace of the product is not the product of traces, that is there exist matrices A and B such that. The trace of the Kronecker product of two matrices is the product of their traces:. Then f and tr are proportional.

Orthogonal matrices - Lecture 7 - Matrix Algebra for Engineers

This is because. If A is symmetric and B is skew-symmetricthen. This leads to generalizations of dimension using trace. The trace of a nilpotent matrix is zero. In particular, using similarity invariance, it follows that the identity matrix is never similar to the commutator of any pair of matrices.

Conversely, any square matrix with zero trace is a linear combinations of the commutators of pairs of matrices. The trace of a Hermitian matrix is real, because the elements on the diagonal are real. The trace of a permutation matrix is the number of fixed pointsbecause the diagonal term a ii is 1 if the i th point is fixed and 0 otherwise.

The trace of a projection matrix is the dimension of the target space. The matrix P X is idempotent, and more generally, the trace of any idempotent matrix equals its own rank.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields.

It only takes a minute to sign up. The trace is the sum of all eigenvalues and the rank is the number of non-zero eigenvalues, which - in this case - is the same thing.

Yes it's true.

Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Relation between trace and rank for projection matrices Ask Question.

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Linked 2. Related 2. Hot Network Questions. Question feed. Mathematics Stack Exchange works best with JavaScript enabled.A Householder matrix is an orthogonal matrix of the form. It is easily verified that is. A Householder matrix is a rank- perturbation of the identity matrix and so all but one of its eigenvalues are.

The eigensystem can be fully described as follows. Fora Householder matrix can be written as.

### Orthogonal Matrix

Simple examples of Householder matrices are obtained by choosingfor which. For we obtain the matrices. Note that the matrix is times a Hadamard matrix. Applying to a vector gives. This equation shows that reflects about the hyperplaneas illustrated in the following diagram, which explains why is sometimes called a Householder reflector. Another way of expressing this property is to writewhere is orthogonal to. Thenso the component of in the direction has been reversed.

If we takethe th unit vector, thenwhich has in the position. In this case premultiplying a vector by flips the sign of the th component. Householder matrices are powerful tools for introducing zeros into vectors. Suppose we are given vectors and and wish to find a Householder matrix such that.

Since is orthogonal, we require thatand since can never equal the identity matrix we also require. But is independent of the scaling ofso we can set.

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Now with we have. Most often we choose to be zero in all but its first component.

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What can we say about square roots of a Householder matrix, that is, matrices such that? We note first that the eigenvalues of are the square roots of those of and so of them will be and one will be. This means that cannot be real, as the nonreal eigenvalues of a real matrix must appear in complex conjugate pairs.

Writewhere is normalized so that. It is natural to look for a square root of the form. Setting leads to the quadratic equationand hence. As expected, these two square roots are complex even though is real. As an example, gives the following square root of the matrix above corresponding to with :.By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service.

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. Can someone write out the proof explicitly please?

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The trace is invariant under cyclic permutations of the argument eg, see this. Hint: Trace is invariant under cyclic permutations, i. Characteristic polynomials are fixed under conjugation, and the trace is one of the coefficients. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Ask Question. Asked 7 years, 7 months ago.

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MathOverflow is a question and answer site for professional mathematicians. It only takes a minute to sign up. This is the socalled "circular real ensemble" of random matrices.

Is it true? Here is a general comment. Some computations reveal that the actual sequence we get is Awhich I've never seen before but which OEIS says are called the Riordan numbers. The sequence begins.

For general compact connected Lie groups the representation graph should look a bit like the intersection of a Weyl chamber with the weight lattice, or something like that. This answer is a follow-up to the other answers particularly to Carlo Beenakker's answer from September 4.

I will just refer to Theorem 3. Here are some random observations and conjectures based on the numerical evidence found by Carlo and that I've dug up elsewhere:. Numerical evidence also suggests the following:. Below I'll mention a classical result due to Alan T.

This quantity was obtained by James using representation theory as is now evident from Qiaochu's more general answer. The reference that on page 40 includes the abovecited result from which I took the above material: Zonal Polynomials by Akimichi Takemura see e. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered.

### Trace (linear algebra)

Moments of the trace of orthogonal matrices Ask Question. Asked 6 years, 1 month ago. Active 10 months ago. Viewed 2k times. Pascoe J. Pascoe 1, 8 8 silver badges 17 17 bronze badges.In linear algebraan orthogonal matrix is a real square matrix whose columns and rows are orthogonal unit vectors orthonormal vectors. This leads to the equivalent characterization: a matrix Q is orthogonal if its transpose is equal to its inverse :. As a linear transformationan orthogonal matrix preserves the inner product of vectors, and therefore acts as an isometry of Euclidean spacesuch as a rotationreflection or rotoreflection.

In other words, it is a unitary transformation. As a linear transformation, every special orthogonal matrix acts as a rotation. An orthogonal matrix is the real specialization of a unitary matrixand thus always a normal matrix. Although we consider only real matrices here, the definition can be used for matrices with entries from any field. However, orthogonal matrices arise naturally from dot productsand for matrices of complex numbers that leads instead to the unitary requirement.

Orthogonal matrices preserve the dot product, [1] so, for vectors u and v in an n -dimensional real Euclidean space. To see the inner product connection, consider a vector v in an n -dimensional real Euclidean space. Written with respect to an orthonormal basis, the squared length of v is v T v. If a linear transformation, in matrix form Q vpreserves vector lengths, then. Thus finite-dimensional linear isometries â€”rotations, reflections, and their combinationsâ€”produce orthogonal matrices.

The converse is also true: orthogonal matrices imply orthogonal transformations. However, linear algebra includes orthogonal transformations between spaces which may be neither finite-dimensional nor of the same dimension, and these have no orthogonal matrix equivalent.

Orthogonal matrices are important for a number of reasons, both theoretical and practical. For example, the point group of a molecule is a subgroup of O 3. Because floating point versions of orthogonal matrices have advantageous properties, they are key to many algorithms in numerical linear algebrasuch as QR decomposition.

As another example, with appropriate normalization the discrete cosine transform used in MP3 compression is represented by an orthogonal matrix. A reflection is its own inversewhich implies that a reflection matrix is symmetric equal to its transpose as well as orthogonal.

The product of two rotation matrices is a rotation matrix, and the product of two reflection matrices is also a rotation matrix. For example. Rotations become more complicated in higher dimensions; they can no longer be completely characterized by one angle, and may affect more than one planar subspace.

Above three dimensions two or more angles are needed, each associated with a plane of rotation. However, we have elementary building blocks for permutations, reflections, and rotations that apply in general. The most elementary permutation is a transposition, obtained from the identity matrix by exchanging two rows. A Householder reflection is constructed from a non-null vector v as. Here the numerator is a symmetric matrix while the denominator is a number, the squared magnitude of v.

This is a reflection in the hyperplane perpendicular to v negating any vector component parallel to v. A Householder reflection is typically used to simultaneously zero the lower part of a column.Each of those contacts will have their own subscriber profile page, with valuable information like social data, member rating, location, activity history, and more.

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