There is an orthogonal matrix when said matrix multiplied by its transpose results in the identity matrix. If the inverse of a matrix is equal to the transpose then the original matrix is orthogonal.
Orthogonal matrices have the characteristic that the number of rows is equal to the number of columns. Furthermore, the row vectors are unit orthogonal vectors and the transpose row vectors are also.
Figure 1. Example of orthogonal matrix and how it transforms geometric objects. (Prepared by Ricardo Pérez)
When an orthogonal matrix is multiplied by the vectors of a vector space, it produces an isometric transformation, that is, a transformation that does not change the distances and preserves the angles.
A typical representative of orthogonal matrices are rotation matrices. The transformations of orthogonal matrices on a vector space are called orthogonal transformations.
The geometric transformations of rotation and reflection of points represented by their Cartesian vectors are carried out by applying orthogonal matrices on the original vectors to obtain the coordinates of the transformed vectors. It is for this reason that orthogonal matrices are widely used in computer graphics processing.
Properties
A matrix M is orthogonal if multiplied by its transpose M T gives as a result the identity matrix I. Similarly, the product of the transpose of an orthogonal matrix by the original matrix results in the identity matrix:
MM T = M T M = I
As a consequence of the previous statement, we have that the transpose of an orthogonal matrix is equal to its inverse matrix:
M T = M -1 .
The set of orthogonal matrices of dimension nxn form the orthogonal group O (n). And the subset of O (n) of orthogonal matrices with determinant +1 form the Group of Unitary Special Matrices SU (n). The matrices of the group SU (n) are matrices that produce linear transformations of rotation, also known as the group of rotations.
Demonstration
We want to show that a matrix is orthogonal if, and only if, the row vectors (or column vectors) are orthogonal to each other and of norm 1.
Suppose that the rows of an orthogonal matrix nxn are n orthonormal vectors of dimension n. If it is denoted by v 1 , v 2 ,…., V n to the n vectors holds:
Where it is evident that indeed the set of row vectors is a set of orthogonal vectors with norm one.
Examples
Example 1
Show that the 2 x 2 matrix that in its first row has the vector v1 = (-1 0) and in its second row the vector v2 = (0 1) is an orthogonal matrix.
Solution: The matrix M is constructed and its transpose M T is calculated:
In this example, the matrix M is self-transposed, that is, the matrix and its transpose are identical. Multiply M by its transpose M T:
It is verified that MM T is equal to the identity matrix:
When the matrix M is multiplied by the coordinates of a vector or a point, new coordinates are obtained that correspond to the transformation that the matrix makes on the vector or point.
Figure 1 shows how M transforms the vector u into u ' and also how M transforms the blue polygon into the red polygon. Since M is orthogonal, it is then an orthogonal transformation, which preserves the distances and the angles.
Example 2
Suppose you have a 2 x 2 matrix defined in the reals given by the following expression:
Find the real values of a, b, c and d such that the matrix M is an orthogonal matrix.
Solution: By definition, a matrix is orthogonal if multiplied by its transpose the identity matrix is obtained. Remembering that the transposed matrix is obtained from the original, exchanging rows for columns, the following equality is obtained:
Performing matrix multiplication we have:
Equating the elements of the left matrix with the elements of the identity matrix on the right, we obtain a system of four equations with four unknowns a, b, c and d.
We propose for a, b, c and d the following expressions in terms of trigonometric ratios sine and cosine:
With this proposal and due to the fundamental trigonometric identity, the first and third equations are automatically satisfied in the equality of the matrix elements. The third and fourth equations are the same and in matrix equality after substituting for the proposed values it looks like this:
which leads to the following solution:
Finally the following solutions are obtained for the orthogonal matrix M:
Note that the first of the solutions has determinant +1 so it belongs to the group SU (2), while the second solution has determinant -1 and therefore does not belong to this group.
Example 3
Given the following matrix, find the values of a and of b so that we have an orthogonal matrix.
Solution: For a given matrix to be orthogonal, the product with its transpose must be the identity matrix. Then, the matrix product of the given matrix with its transposed matrix is carried out, giving the following result:
Next, the result is equated with the 3 x 3 identity matrix:
In the second row, the third column has (ab = 0), but a cannot be zero, because otherwise the equality of the elements in the second row and second column would not be fulfilled. Then necessarily b = 0. Substituting b for the value 0 we have:
Then the equation is solved: 2a ^ 2 = 1, whose solutions are: + ½√2 and -½√2.
Taking the positive solution for a, the following orthogonal matrix is obtained:
The reader can easily verify that the row vectors (and also the column vectors) are orthogonal and unitary, that is, orthonormal.
Example 4
Show that the matrix A whose row vectors are v1 = (0, -1 0), v2 = (1, 0, 0) and v3 = (0 0 -1) is an orthogonal matrix. Additionally find the vectors are transformed from the canonical basis i, j, k to vectors u1, u2 and u3.
Solution: It should be remembered that the element (i, j) of a matrix multiplied by its transpose, is the dot product of the vector of row (i) by that of column (j) of the transpose. Furthermore, this product is equal to the Kronecker delta in the case that the matrix is orthogonal:
In our case it looks like this:
v1 • v1 = 0x0 + (-1) x (-1) + 0x0 = 1
v2 • v2 = 1 × 1 + 0x0 + 0x0 = 1
v3 • v3 = 0x0 + 0x0 + (-1) x (-1) = 1
v1 • v2 = 0x1 + (-1) x0 + 0x0 = 0
v2 • v1 = 1 × 0 + 0x (-1) + 0x0 = 0
v2 • v3 = 1 × 0 + 0x (0) + 0x (-1) = 0
v3 • v2 = 0x1 + 0x (0) + (-1) x0 = 0
v1 • v3 = 0x0 + (-1) x (0) + 0x (-1) = 0
v3 • v1 = 0x0 + 0x (-1) + (-1) x0 = 0
With which it is shown that it is an orthogonal matrix.
Furthermore u1 = A i = (0, 1, 0); u2 = A j = (-1, 0, 0) and finally u3 = A k = (0, 0, -1)
References
- Anthony Nicolaides (1994) Determinants & Matrices. Pass Publication.
- Birkhoff and MacLane. (1980). Modern Algebra, ed. Vicens-Vives, Madrid.
- Casteleiro Villalba M. (2004) Introduction to linear algebra. ESIC Editorial.
- Dave Kirkby (2004) Maths Connect. Heinemann.
- Jenny Olive (1998) Maths: A Student's Survival Guide. Cambridge University Press.
- Richard J. Brown (2012) 30-Second Maths: The 50 Most Mind-Expanding Theories in Mathematics. Ivy Press Limited.
- Wikipedia. Orthogonal matrix. Recovered from: es.wikipedia.com
- Wikipedia. Orthogonal matrix. Recovered from: en.wikipedia.com