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Problem 621

Apply Gershgorin's theorem to estimate the eigenvalues of the matrix $$ \begin{array}{rlrr} & \mid(5 / 2) & -(1 / 2) & 1 \\ \mathrm{~A}= & (3 / 2) & (1 / 2) & -1 \mid \\ & 0 & 0 & 0 \end{array} $$

Short Answer

Expert verified
Based on Gershgorin's theorem, the estimated eigenvalue intervals of the matrix \(A = \begin{pmatrix} \frac{5}{2} & -\frac{1}{2} & 1 \\ \frac{3}{2} & \frac{1}{2} & -1 \\ 0 & 0 & 0 \end{pmatrix}\) are \(I_1 = [1,4]\) and \(I_2 = [-2,3]\). Therefore, the possible intervals for the eigenvalues are \(I_1 \cup I_2 = [-2,4]\).
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Step 1: 1. Identify the matrix A

First, we need to rewrite the given matrix $$A$$ in a clear form: $$ A = \begin{pmatrix} \frac{5}{2} & -\frac{1}{2} & 1 \\ \frac{3}{2} & \frac{1}{2} & -1 \\ 0 & 0 & 0 \end{pmatrix} $$

Step 2: 2. Construct the Gershgorin circles

To construct the Gershgorin circles, we need to find the radius for each circle based on the sum of the absolute values of the off-diagonal elements in each row. Radius of the first circle (corresponding to the first row): $$ R_1 = |-\frac{1}{2}| + |1| = \frac{1}{2} + 1 = \frac{3}{2} $$ Radius of the second circle (corresponding to the second row): $$ R_2 = |\frac{3}{2}| + |-1| = \frac{3}{2} + 1 = \frac{5}{2} $$ Since the third row contains only zeros, there is no associated circle, and we can omit it from our analysis.

Step 3: 3. Estimate the eigenvalue intervals

Now, we will find the intervals for the eigenvalues using the Gershgorin circles. The eigenvalues are within the union of these intervals. Eigenvalue interval for the first circle (centered at \(5/2\)): $$ I_1 = [\frac{5}{2} - \frac{3}{2}, \frac{5}{2} + \frac{3}{2}] = [1, 4] $$ Eigenvalue interval for the second circle (centered at \(1/2\)): $$ I_2 = [\frac{1}{2} - \frac{5}{2}, \frac{1}{2} + \frac{5}{2}] = [-2, 3] $$ So, the estimated eigenvalue intervals are \(I_1 \cup I_2 = [-2, 4]\). Keep in mind that these are only estimates, and the actual eigenvalues may be different, although they will belong to the given intervals.

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