Determine whether the matrix is orthogonal, and if so find it inverse. (a) [ 0 1 (1)/

Determine whether the matrix is orthogonal, and if so find...

Key Concepts

Orthogonal Matrix

An orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors. This means that each column (and each row) has unit length and is perpendicular to all other columns (or rows). One of the key properties of an orthogonal matrix is that its inverse is simply its transpose, making it computationally efficient to work with in many applications related to rotations and reflections in space.

Orthonormal Vectors

Orthonormal vectors are vectors that are both orthogonal (perpendicular) to each other and of unit length. In the context of matrices, if the columns (or rows) of a matrix form an orthonormal set, then the matrix is orthogonal. The inner product (dot product) of distinct vectors in this set is zero, and the inner product of a vector with itself equals one, ensuring the orthonormality condition.

Matrix Transpose

The transpose of a matrix is an operation that flips the matrix over its diagonal, converting the rows to columns and vice versa. In the context of orthogonal matrices, this operation is especially significant since the transpose of an orthogonal matrix is equal to its inverse. This property is utilized to verify the orthogonality of a given matrix and simplify inversion computations.

Matrix Inverse

The inverse of a square matrix is another matrix such that when it is multiplied with the original matrix, the result is the identity matrix. For orthogonal matrices, calculating the inverse is particularly straightforward because the inverse is the transpose of the matrix. This property not only simplifies the computation but also reinforces the role of orthogonal matrices in various transformations, including rotations and reflections.

Transcript

00:01 Hi, so for this exercise we have this matrix a and we need to determine if this matrix is orthogonal or not and there are several ways where you can verify the matrix is orthogonal or not and just to remind you that one of the properties that orthogonal matrices have is that the transpose is the inverse of this matrix that means that this kind of matrices satisfy this condition a times it transpose is equal to the identity so in this sense we can take calculate the transpose of this matrix and then multiply the matrix and after it identity or there are other ways to verify this that sometimes could be easier to calculate and is that to ber it consists and verify that the column space or the columns of this matrix form an orthomormal set so in this case this is going to be the first vector is going to be the vector u2 and this will be the vector u3 so we need to verify that these three vectors form an orthonormal set so let's check if they are orthogonal between them so u1 with u2 is clearly equals to zero being their product u1 with u3 is also equals to and then u2 with u3 and here happens something in this that the inner product of these two vectors is going to be equals to 1 over the square root of 2 which clearly is different from 0 so they don't form an orthomormal set so this set is not orthonormal and therefore the matrix a is not orthogonal and then we have another case of another matrix so we need to here we are going to repeat the procedure so we are going to consider u1 you 2 and u3 and let's calculate the corresponding inner products so u1 with u2 it's going to be equals of 1 over the square root of 12 times minus 1 plus 1 which is equal to 0 the inner product of 1 with u 3 this equals to 1 over the square root of 6 times minus 1 plus 1 which again is equal to 0 then the last one is the inner product of 2 with you 3 which in this case is equal to 1 over 3 times the square root of 2 times 1 minus 2 plus 1 which again is equal to 0 so so far this 3 vectors form an orthogonal set but now we need to verify that they are an orthonormal set and that means that the the length of each of these vectors is equal to one in other words each vector is unitary so for you one this is clear this is equal to one for u2 this is equal to the square root of 36 over 36 which is equal to one and the sorry here is 6 over 6 and the length of the norm of you 3 is equal to the square root of 3 over 3 which again is equal to 1 so the 3 vectors have length 1 or norm 1 and that means that they form an orthomormal set and if the columns of a matrix form an orthomormal set that implies that the matrix a is orthogonal.

04:20 Now we know that the matrix a is orthogonal and then we can easily calculate the inverse of this matrix...

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