Understanding Orthogonal and Unitary Matrices
Learn about orthogonal and unitary matrices, their definitions, properties, and a problem-solving example. Discover how to determine if a matrix is singular based on the determinants of orthogonal matrices. Explore the concepts with informative images and explanations.
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Presentation Transcript
Definition A real n n matrix A is said to be orthogonal if AAt=In. A complex n n matrix A is said to be unitary if AA0=In.
Examples 2/3 1/ 2 1/3 2 1/3 0 2/3 1/ 2 1/3 2 is a orthogonal matrix. ? = 4/3 2 A=(1 + ?)/2 (1 ?)/2 (1 ?)/2 (1 + ?)/2is a unitary matrix.
Properties of orthogonal matrix 1.If A be an orthogonal matrix then A-1=At. 2.If A and B be orthogonal matrices of the same order then AB is orthogonal. 3.If A be an orthogonal matrix ,then A-1is orthogonal.
Properties of Unitary matrix If A be a unitary matrix then A is non singular and modulus of det A=1 If A be a unitary matrix then A-1=A0
Problem Q. A and B are real orthogonal matrices of the same order and detA +detB=0.Show that A+B is a singular matrix. Solution-we have detA=-detB, AAt=In,BBt=In. det(A+B)=det A(In+AtB)=detA(Bt+At)B=-det(A+B)t=-det(A+B) i.e det(A+B)=0 Hence A+B is singular matrix .
