How can I compute the average eigenvalue of a parametrised matrix? I have a family of n times n parameterised Hermitian matrices, the simplest of which is of the form: M(t)=f(t)I+Q.

Maverick Avery

Maverick Avery

Answered question

2022-10-20

How can I compute the average eigenvalue of a parametrised matrix?
I have a family of n × n parameterised Hermitian matrices, the simplest of which is of the form:
M ( t ) = f ( t ) I + Q
where f(t) is a polynomial in t, I is the usual n × n identity matrix and Q is a known n × n Hermitian matrix. I need to compute the (arithmetic) mean of each of the eigenvalues { λ α ( t ) } of M(t) over an interval a t b. What is the best way to do this?

Answer & Explanation

Marlene Welch

Marlene Welch

Beginner2022-10-21Added 23 answers

Step 1
Eigenvalues of matrix M are the roots of det ( M λ I ) = 0. Or in your case:
det ( Q + f ( t ) I λ I ) = det ( Q ( λ f ( t ) I ) .
Step 2
Thus, if λ ( 0 ) is an eigenvalue at t = 0, then λ ( τ ) = λ ( 0 ) + f ( τ ) f ( 0 ) is an eigenvalue at t = τ.
bergvolk0k

bergvolk0k

Beginner2022-10-22Added 4 answers

Step 1
If Q x i = λ i x i then M ( t ) x i = ( f ( t ) + λ i ) x i .
Step 2
So, λ ¯ i ( M ) = λ i + 1 b a a b f ( τ ) d τ

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