Orthogonal Matrix

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• To show a given matrix is orthogonal

  is orthogonal. ------------------------------------------------------------------------------------ A matrix is shown to be orthogonal.

• Find a unitary matrix which is not orthogonal and an orthogonal matrix which is not unitary.

  Find a unitary matrix which is not orthogonal and an orthogonal matrix which is not unitary.

• Matrix Reflection and Rotation Problems

  is orthogonal A matrix A is an orthogonal matrix if , where is the transpose of A and I is the identity matrix.

• Properties of Determinants : Orthogonal Matrix and Parallelepiped

  37132 Properties of Determinants: Orthogonal Matrix and Parallelepiped If Q is an orthogonal matrix, so that QTQ = I, prove that det Q equals +1 or -1. What kind of parallelepiped is formed from the rows (or columns) of an orthogonal matrix Q?

• Matrix Representation : Orthogonal Transformation and Determinant

  Write a proof for the following statement: If P is an n x n orthogonal matrix, then det(P) = 1 or -1. Show work. Proof. By the definition of orthogonal matrix, we know that , where E is the identity matrix of order n.

• Orthogonal Matrix and Eigenvalues : Let A be a 3 x 3 real orthogonal matrix with det A = 1. Prove that λ= 1 is an eigenvalue of A.

  64737 Orthogonal Matrix and Eigenvalues : Let A be a 3 x 3 real orthogonal matrix with det A = 1. Prove that λ= 1 is an eigenvalue of A. Let A be a 3 x 3 real orthogonal matrix with det A = 1. Prove that λ= 1 is an eigenvalue of A.

• Orthogonal projection

  2649 Orthogonal Projection Matrix Theory See attached file for full problem description with symbols and equations. --- Definition 11.1 An orthogonal projection operator is a linear transformation such that and .

• Properties of condition numbers : Orthogonal Matrices and Eigenvalues

  If is a multiple of an orthogonal matrix, we know for each orthogonal matrix , we have . So and thus . Hence we have . If , from 9, we know . This implies that all the eigenvalues of the symmetric matrix are the same. So .

• Orthogonal basis to find matrix

  This shows how to construct a given matrix using an orthogonal basis.

• Matrix Representation : Orthogonal Transformation

  2 0 ] is an orthogonal matrix.