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I checked weather the following matrix is diagonalizable.

$$A=\begin{bmatrix} 4 & 0 & 4\\ 0 & 4 & 4\\ 4 & 4 & 8 \end{bmatrix}$$

And the corresponding eigenvalues were $0$, $4$, $12$.

Now if we write the similar diagonal matrix to $A$ it would be,

$$D=\begin{bmatrix} 0 & 0 & 0\\ 0 & 4 & 0\\ 0 & 0 & 12 \end{bmatrix}$$

which is theoretically not a diagonal matrix.

Now is $A$ diagonalizable?

  • 11
    Why do you think $D$ is not a diagonal matrix? – 2017-01-05
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    A matrix is diagonal if every coefficient off the diagonal is $0$; nothing at all is imposed on diagonal coefficients. – 2017-01-05
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    @RobertIsrael My bad. I thought diagonal matrix means all the diagonal entries should be non zero. – 2017-04-20

3 Answers 3

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$D$ is a diagonal matrix. A square matrix is a diagonal matrix if and only if the off-diagonal entries are $0$.

Hence your matrix is diagonalizable.

In fact, if the eigenvalues are all distinct, then it is diagonalizable.

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    Moreover, every real symmetric matrix is diagonalizable, whether or not its eigenvalues are distinct. – 2017-01-05
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    Excellent point! – 2017-01-05
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Every Matrix is diagonalisable if it's eigenvalues are all distinct, no matter the values of the eigenvalue theirselves. Also your second Matrix is diagonal, cause you only have terms on the diagonal, respectively 0, 4 and 12.

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Would the fact that every real symmetric matrix is diagonalizable come immediately from the fact that given a real inner product space and a self adjoint operator then it has a orthonormal basis of eigenvectors?