- #1
rjw5002
Homework Statement
Suppose [tex]P \in L(V)[/tex] and P^2 = P. Prove that (I+P) is invertible.
Homework Equations
The Attempt at a Solution
Am I right to assume that since P^2 = P, P = I?
Don't give out answers. :tongue:Dick said:Nope. If P=[[1,0],[0,0]], P^2=P. Is that all you wanted to know? Try solving (I+P).(I+aP)=I for a real number a. If you can solve it, you've found an inverse.
That doesn't even work for real numbers: x^2 = x has two solutions. (What are they? How would you find them?)rjw5002 said:Am I right to assume that since P^2 = P, P = I?
The problem, IMHO, is that there is nothing left of the original problem -- you've simply presented him with an entirely different (and much easier) problem, and absolutely no motivation why someone would ever think of the new problem.Dick said:Ok, I'll admit, that's a little close to giving the answer for comfort. I thought about stopping after the '?'. But that seemed a little too close to just teasing the OP.
Why?rjw5002 said:Using the fact that P^2 = P, we can say that P is a diagonal matrix.
The original poster is on one right track, I think. The problem is easy if the matrix is diagonal, and if he can prove the problem can be reduced to the diagonal case, he's done. I don't remember how easy that is, though.Dick said:I, in fact, solved it by ansatz. I guessed a form for the solution which seemed to have enough parameters to be solvable given powers of P reduce to linear functions of P. That's all. If you have a better motivation, go for it. Suppose I could have explained that, though.
Hurkyl said:The original poster is on one right track, I think. The problem is easy if the matrix is diagonal, and if he can prove the problem can be reduced to the diagonal case, he's done. I don't remember how easy that is, though.
I'm a little disappointed that my favorite trick of expanding 1/(1+P) as a power series doesn't quite work; it requires you to do a deformation to arrive at the correct answer. Though this would give one method of motivating your approach.
Another method of motivating your approach would be to invoke the knowledge that matrices have minimal polynomials, and there is an easy way to extract the inverse of your matrix from its minimal polynomial.
Of course, you could simply guess at the form of the answer, as you did, but I feel like simply writing the form of the answer doesn't really help the OP learn.
P and 1-P are complementary projection matrices.Dick said:I don't THINK P is necessarily diagonalizable. It might be but I don't see what compels it to be.
I'm hoping so.Hopefully the OP will learn from this exchange as well.
An identity matrix is a special type of square matrix where the values along the main diagonal are 1 and all other values are 0. It is denoted by the symbol I and has the property that when multiplied by any other matrix, it returns that same matrix as the product.
The identity matrix is used as a neutral element in matrix operations. It is similar to the number 1 in multiplication and the number 0 in addition. When a matrix is multiplied by the identity matrix, the result is the original matrix, making it a useful tool in solving equations and performing transformations.
An inverse matrix is a matrix that, when multiplied by another matrix, results in the identity matrix. It is denoted by the symbol A-1 and has the property that A x A-1 = I. In other words, it "undoes" the original matrix and is used to solve equations and perform transformations.
To find the inverse of a matrix, you can use the Gauss-Jordan elimination method or the adjugate method. In the Gauss-Jordan method, the matrix is transformed into reduced row echelon form and the inverse is found by using elementary row operations. In the adjugate method, the inverse is found by calculating the adjugate matrix and dividing it by the determinant of the original matrix. There are also online calculators and software programs that can find the inverse of a matrix for you.
Finding the inverse of a matrix is important in various applications in mathematics, physics, and engineering. It is used to solve systems of linear equations, find the inverse of a transformation, and perform operations such as matrix division. It is also a key concept in linear algebra and plays a crucial role in understanding and solving many problems in these fields.