Adding a constant times the unit matrix and eigenvalues

In summary, to find the eigenvalues of a matrix A, you solve the equation det |A - \lambda I| = 0. When a constant e is added to the matrix A, the equation becomes det |(A + eI) - \lambda_{new} I| = 0, which is equivalent to solving det |A - (\lambda_{new} - e) I| = 0. This implies that \lambda_{new} = \lambda + e. Additionally, the eigenvectors of A and (A+eI) coincide.
  • #1
julian
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To find the eigenvalues [itex]\lambda[/itex] of a matrix [itex]A[/itex] you solve the equation

[itex]det |A - \lambda I| = 0[/itex] eq(1)

but now what if you add [itex]e I[/itex] to the matrix A where e is a constant? Then you have to solve the equation,

[itex]det |(A + eI) - \lambda_{new} I| = 0[/itex] eq(2)

which is the same as solving

[itex]det |A - (\lambda_{new} - e) I| = 0[/itex] eq(3)

Doesn't comparison of eq(3) with eq(1) just imply [itex]\lambda_{new} = \lambda + e[/itex]?
 
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  • #2
look at it this way [itex](A+eI)v=λ_{new}v[/itex] for an eigenvector v of A+eI.

Now suppose w is an eigenvector of A.

then [itex](A+eI)w=λw+ew=(λ+e)w[/itex]

so in fact even the eigenvectors coincide.
 
  • #3
Thanks cus a PhD in maths didn't know what I was talking about...I was just checking I was right. Your way is more transparent.
 

FAQ: Adding a constant times the unit matrix and eigenvalues

What does it mean to add a constant times the unit matrix to a matrix?

When we add a constant times the unit matrix to a matrix, we are essentially adding a multiple of the identity matrix to the original matrix. This means that each element in the original matrix is multiplied by the constant and added to the corresponding element in the identity matrix. This operation is often used in linear algebra to make matrices easier to work with or to transform them in some way.

2. How does adding a constant times the unit matrix affect the eigenvalues of a matrix?

When we add a constant times the unit matrix to a matrix, the eigenvalues of the original matrix do not change. This is because the eigenvalues are determined by the characteristics of the original matrix itself, not by any operations we perform on it. However, adding a constant times the unit matrix can change the eigenvectors of the matrix, which in turn can change how the matrix behaves in certain operations.

3. Can we add a constant times the unit matrix to any matrix?

Yes, we can add a constant times the unit matrix to any matrix as long as the dimensions of the matrices are compatible. This means that the number of rows and columns in the two matrices must be the same. If the dimensions are not compatible, we cannot perform the addition operation.

4. What is the purpose of adding a constant times the unit matrix to a matrix?

Adding a constant times the unit matrix to a matrix can serve several purposes, depending on the context. In some cases, it may be used to simplify a matrix or make it easier to work with. In other cases, it may be used to transform the matrix in some way, such as scaling or shifting its values. It can also be used in certain calculations or algorithms in linear algebra.

5. Are there any other operations similar to adding a constant times the unit matrix?

Yes, there are other operations that are similar to adding a constant times the unit matrix, such as multiplying a matrix by a scalar (a single number) or adding/subtracting another matrix. These operations also involve performing a mathematical operation on each element in the matrix, but they may have different effects on the matrix's values and properties.

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