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If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal?
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If A is an orthagonal matrix and B=AP where P is non singular matrix t...
Proof:
Step 1: Show that P is also orthogonal
Since A is orthogonal, we know that:
AA^T = A^TA = I
We can rewrite B as:
B = AP = A(P^T)^T
So, we have:
BB^T = A(P^T)(P^T)^TA^T
Since P is non-singular, we know that P^T is also non-singular. Therefore, we can multiply both sides by (P^T)^-1:
BB^T(P^T)^-1 = AA^T
Since A is orthogonal, we know that AA^T = I. Therefore, we have:
BB^T(P^T)^-1 = I
Multiplying both sides by B^-1, we get:
B^-1BB^T(P^T)^-1 = B^-1
Since B = AP, we can substitute to get:
P^T(A^TA)(P^T)^-1 = B^-1
Simplifying, we get:
P^TP = B^-1
Therefore, we have shown that P is also orthogonal.
Step 2: Show that PB^-1 is orthogonal
We can rewrite PB^-1 as:
PB^-1 = A(P^T)^T(B^-1)^T
Multiplying by its transpose, we get:
(PB^-1)(PB^-1)^T = A(P^T)^T(B^-1)^T(B^-1)(P^T)A^T
Since P is orthogonal, we know that P^T = P^-1. Therefore, we can simplify to get:
(PB^-1)(PB^-1)^T = A(P^-1)(B^-1)(B^-1)(P^-1)A^T
Simplifying, we get:
(PB^-1)(PB^-1)^T = AA^T
Since A is orthogonal, we know that AA^T = I. Therefore, we have:
(PB^-1)(PB^-1)^T = I
Therefore, we have shown that PB^-1 is also orthogonal.
Community Answer
If A is an orthagonal matrix and B=AP where P is non singular matrix t...
A linear transformation T : Rn → Rn is orthogonal if and only if T preserves the dot product: ... Theorem 6 (5.3.4, products and inverses of orthogonal matrices). a) The product AB of two orthogonal n × n matrices A and B is orthogonal. b) The inverse A−1 of an orthogonal n × n matrix A is orthogonal.
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If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal?
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If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal?.
If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If A is an orthagonal matrix and B=AP where P is non singular matrix then show that the matrix PB^-1 is also orthagonal?.
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