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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?.
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