Trace of a matrix equals sum of its determinants?

Thread starter samspotting
Start date Feb 8, 2009
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Determinants Matrix Sum Trace

In summary, the trace of a matrix is the sum of its main diagonal elements and can be calculated by adding them up. The determinant of a matrix is a value that can give information about the matrix's properties. When the trace equals the sum of the determinants, it satisfies the matrix trace theorem. The trace is important in linear algebra and has applications in physics and engineering.

Feb 8, 2009

samspotting

If a matrix is diagonalizable, how does its trace equal the sum of its eigenvalues?

I can't find a proof for this anywhere.

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Feb 8, 2009

Hurkyl

Staff Emeritus

Science Advisor

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What is the trace of a diagonal matrix? What are the algebraic properties of trace?

Feb 9, 2009

samspotting

Ah I see, the trace is similarity invariant. thanks

1. What is the trace of a matrix?

The trace of a matrix is the sum of the elements on its main diagonal. In other words, it is the sum of the values in the top left to bottom right direction.

2. What is the determinant of a matrix?

The determinant of a matrix is a value that can be calculated from the elements of the matrix. It gives information about the matrix such as whether it is invertible or singular.

3. How do you calculate the trace of a matrix?

To calculate the trace of a matrix, simply add up the elements on the main diagonal. For example, if a matrix is [1 23 4], the trace would be 1+4=5.

4. What does it mean when the trace of a matrix equals the sum of its determinants?

When the trace of a matrix is equal to the sum of its determinants, it means that the matrix satisfies the matrix trace theorem. This theorem states that the sum of the determinants of all submatrices of a matrix is equal to the trace of the matrix.

5. Why is the trace of a matrix important?

The trace of a matrix is important because it is a useful tool in linear algebra. It can provide information about the matrix, such as its eigenvalues, and can also be used to simplify calculations and proofs. Additionally, the trace of a matrix has applications in fields such as physics and engineering.