Finding unitary transformation

In summary, the conversation discusses the task of finding a unitary transformation to diagonalize a given matrix. The approach to this problem is to first find the eigenvalues, but the speaker notes that there may be a way to reduce the matrix beforehand. However, they are unsure of the best method for this and suggests finding eigenvectors through inspection and guessing. Ultimately, the speaker advises to evaluate the determinant to find the remaining two eigenvectors.
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Homework Statement



Find a unitary transformation that diagonalizes the matrix:

1 1 1 -3
1 1 1 -3
1 1 1 -3
-3 -3 -3 -9


Homework Equations





The Attempt at a Solution


So before I even start with finding the eigenvalues for this, I know there has to be a way to reduce this so I don't have to find the determinant of a 4x4 matrix. Clearly none of these rows are independent. We haven't gone over this very thoroughly in class, so I'm not too sure about the best way to go about this.
 
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  • #2
There's no real shortcut here. Sometimes you can find eigenvectors by inspection and guessing. Two of them are pretty easy to find. I'm not how you would guess the other two, unless you are better at guessing than I am. Just evaluate the determinant.
 

1. What is a unitary transformation?

A unitary transformation is a mathematical operation that preserves the length and angle between vectors. In simpler terms, it is a transformation that does not change the shape or size of an object.

2. Why is it important to find unitary transformations?

Unitary transformations are important in many areas of science, particularly in quantum mechanics and linear algebra. They allow us to analyze and manipulate complex systems by simplifying them into simpler unitary operations.

3. How do you find a unitary transformation?

To find a unitary transformation, you can use several methods such as diagonalization, Gram-Schmidt process, or singular value decomposition. These methods involve finding a set of orthogonal vectors that form a basis for the transformation matrix.

4. Can any matrix be transformed into a unitary matrix?

No, not all matrices can be transformed into a unitary matrix. Only square matrices with orthogonal columns can be transformed into unitary matrices. The transformation process also depends on the properties of the matrix, such as its eigenvalues and eigenvectors.

5. How is a unitary transformation useful in quantum mechanics?

In quantum mechanics, unitary transformations are used to describe the evolution of systems over time. They are used to calculate the probabilities of different states of a quantum system and to transform the states of particles when they interact with each other.

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