Question

If two eigen values of \(\left[\begin{array}{rrr}8 & -6 & 2 \\ -6 & 7 & -4 \\\ 2 & -4 & 3\end{array}\right]\) are 3 and 15, then the third eigen value is

Short answer

The third eigenvalue is 0.

Step-by-step solution

Understand the Properties of Eigenvalues

Eigenvalues of a matrix are solutions to the characteristic equation, which is derived from the determinant of the matrix minus lambda times the identity matrix equals zero. An important property of eigenvalues is that the sum of the eigenvalues of a matrix equals the trace of the matrix.

Calculate the Trace of the Given Matrix

The trace of a matrix is the sum of the elements along its main diagonal. For the given matrix, the trace is obtained by adding 8, 7, and 3, which gives us 18. Therefore, \( \text{trace} = 8 + 7 + 3 = 18 \).

Use the Sum of Eigenvalues Property

Since the sum of the eigenvalues is equal to the trace of the matrix, for this 3x3 matrix, we have: \( \lambda_1 + \lambda_2 + \lambda_3 = 18 \). We know two eigenvalues (\(\lambda_1 = 3\) and \(\lambda_2 = 15\)), so the equation becomes: \(3 + 15 + \lambda_3 = 18 \).

Solve for the Third Eigenvalue

With the simplified equation \(3 + 15 + \lambda_3 = 18\), solve for \(\lambda_3\) by subtracting the sum of known eigenvalues from the trace: \( \lambda_3 = 18 - (3 + 15) = 18 - 18 = 0 \).

Key concepts

Trace of a Matrix

The trace of a matrix is a straightforward yet crucial concept that aids in understanding eigenvalues. In the context of our given matrix,

• The trace is simply the sum of the elements found on the matrix's main diagonal.
• These diagonal elements play a significant role, particularly when finding eigenvalues.

For our specific 3x3 matrix: \[\begin{array}{ccc} 8 & -6 & 2 \-6 & 7 & -4 \2 & -4 & 3 \end{array}\]the main diagonal consists of the numbers 8, 7, and 3.
Adding these gives us the trace: \( 8 + 7 + 3 = 18 \).
Remember, the trace reflects certain properties of the matrix, like how it relates to the sum of the eigenvalues. Understanding this connection simplifies many complex problems related to matrices.

Characteristic Equation

The characteristic equation is essential for finding the eigenvalues of any matrix. It emerges from a specific process:

• First, you subtract \( \lambda \) times the identity matrix from the original matrix. This helps isolate the impact of each diagonal element against \( \lambda \).
• Next, you compute the determinant of this resulting matrix.
• Finally, set this determinant equal to zero. This forms our characteristic equation.

By solving the characteristic equation, you identify the lambda (\( \lambda \)) values that are the eigenvalues of the matrix.
For example, with our given matrix, solving the characteristic equation lets us confirm known eigenvalues of 3 and 15, and helps in identifying additional ones.
The elegance of this process lies in its ability to transform the potentially complex interactions within a matrix into a manageable equation that highlights key properties.

Sum of Eigenvalues Property

The sum of eigenvalues property is an elegant theorem with practical applications in matrix analysis. This principle states that:

• The sum of all eigenvalues of a matrix equals the trace of the matrix.
• This easy-to-remember rule holds true for any square matrix.
• With this property, you can confidently trace eigenvalues and fill in missing pieces of the puzzle.

In our example, we know the trace of the matrix is 18. Given eigenvalues 3 and 15,
we apply the sum of eigenvalues property: \( \lambda_1 + \lambda_2 + \lambda_3 = 18 \).
Substituting known values, we find the third eigenvalue: \( \lambda_3 = 18 - (3 + 15) = 0 \).
Recognizing how one simple property can so easily resolve matrix queries helps make sense of the remarkable structures in mathematical arrays.