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Snowcat [4.5K]
3 years ago
6

In general, the eigenvalues of a diagonal matrix are given by the entries on the diagonal. Verify this for 2 × 2 and 3 × 3 matri

ces.
Mathematics
1 answer:
egoroff_w [7]3 years ago
7 0

Eigenvalues:

Eigenvalues of a matrix A are all the values of π for the following equation:

d = det (πI-A) = 0

For a 2x2 diagonal matrix:

Let A =\left[\begin{array}{ccc}a1&0\\0&a2\\\end{array}\right] \\and\\I=\left[\begin{array}{ccc}1&0\\0&1\\\end{array}\right]

\pi I-A=\left[\begin{array}{ccc}\pi-a1&0\\0&\pi-a2\\\end{array}\right] \\

Now,

d = det (πI-A)

π1-a1=0 and π2-a2=0

a1=π1 and a2=π2

Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.

For a 3x3 diagonal matrix:

Let A =\left[\begin{array}{ccc}a1&0&0\\0&a2&0\\0&0&a3\end{array}\right] \\and\\I=\left[\begin{array}{ccc}1&0&0\\0&1&0\\0&0&1\end{array}\right]

\pi I-A=\left[\begin{array}{ccc}\pi-a1&0&0\\0&\pi-a2\\0&0&\pi-a3\end{array}\right] \\

Now,

d = det (πI-A)

π1-a1=0  ;  π2-a2=0  ;  π3-a3=0  

a1=π1   ;   a2=π2   ;   a3=π3

Hence proved that the eigenvalues of a diagonal matrix are given by the entries on the diagonal.

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