I found out that there exist positive definite matrices that are non-symmetric, and I know that symmetric positive definite matrices have positive eigenvalues.

Does this hold for non-symmetric matrices as well?

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    $\begingroup$ Caution: there is no general agreement on what "positive definite" means for non-Hermitian matrices. Which definition are you using? $\endgroup$ Nov 17 '11 at 18:28
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    $\begingroup$ How about: $$[x\ y]\left[\matrix{1&1\cr -1&1}\right]\left[\matrix{x \cr y}\right]=[x\ y]\left[\matrix {x+y\cr-x+y}\right]=(x^2+xy)+(-xy+y^2)=x^2+y^2,$$ and $$ \left|\matrix {1-\lambda &1\cr -1&1-\lambda } \right| =(1-\lambda)^2+1\ne 0. $$ $\endgroup$ Nov 17 '11 at 18:42
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    $\begingroup$ I've said it before and I'll say it again: positive-definite should not be a term that applies to matrices. It should only apply to quadratic forms, which are naturally described by symmetric matrices only. $\endgroup$ Nov 17 '11 at 18:46
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    $\begingroup$ It's a nice sentiment, but the genie's out of the bottle. $\endgroup$ May 18 '13 at 21:08
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    $\begingroup$ There is a nice explanation about non-hermitian positive definite matrices. Please have a look into math.technion.ac.il/iic/ela/ela-articles/articles/… $\endgroup$
    – user225624
    Mar 22 '15 at 19:43

Let $A \in M_{n}(\mathbb{R})$ be any non-symmetric $n\times n$ matrix but "positive definite" in the sense that:

$$\forall x \in \mathbb{R}^n, x \ne 0 \implies x^T A x > 0$$ The eigenvalues of $A$ need not be positive. For an example, the matrix in David's comment:


has eigenvalue $1 \pm i$. However, the real part of any eigenvalue $\lambda$ of $A$ is always positive.

Let $\lambda = \mu + i\nu\in\mathbb C $ where $\mu, \nu \in \mathbb{R}$ be an eigenvalue of $A$. Let $z \in \mathbb{C}^n$ be a right eigenvector associated with $\lambda$. Decompose $z$ as $x + iy$ where $x, y \in \mathbb{R}^n$.

$$(A - \lambda) z = 0 \implies \left((A - \mu) - i\nu\right)(x + iy) = 0 \implies \begin{cases}(A-\mu) x + \nu y = 0\\(A - \mu) y - \nu x = 0\end{cases}$$ This implies

$$x^T(A-\mu)x + y^T(A-\mu)y = \nu (y^T x - x^T y) = 0$$

and hence $$\mu = \frac{x^TA x + y^TAy}{x^Tx + y^Ty} > 0$$

In particular, this means any real eigenvalue $\lambda$ of $A$ is positive.

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    $\begingroup$ Is the converse true? If all of the eigenvalues of a matrix $A$ have positive real parts, does this mean that $x^TAx > 0$ for any $x \ne 0 \in \mathbb{R}^n$? What if we assume $A$ is diagonalizable? $\endgroup$
    – nukeguy
    Feb 15 '17 at 18:24
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    $\begingroup$ @nukeguy The converse is false. example $A = \begin{bmatrix}3 & 7\\1 & 3\end{bmatrix}$ and $x = \begin{bmatrix}1\\-1\end{bmatrix}$. $\endgroup$ Feb 16 '17 at 4:40
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    $\begingroup$ Can we put an extra condition (perhaps a bound on the off-diagonal entries in terms of the diagonal entries?) that in addition to a positive spectrum guarantees positivity of the matrix? $\endgroup$ Jan 23 '19 at 19:32
  • $\begingroup$ @achillehui Nice answer, and upvoted: Does the counter example matrix you gave above have a square root? $\endgroup$ Nov 19 '19 at 14:29
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    $\begingroup$ @Mathmath, yes, $A$ do have square roots, e.g. $ \frac{\sqrt{2}I_2+A}{\sqrt{6 + 2\sqrt{2}}}$ is one such square root. $\endgroup$ Nov 19 '19 at 14:54

I am answering the first part of @nukeguy's comment, who asked

``Is the converse true? If all of the eigenvalues of a matrix $𝐴$ have positive real parts, does this mean that $π‘₯^𝑇𝐴π‘₯>0$ for any $π‘₯β‰ 0βˆˆβ„^𝑛$? What if we assume $𝐴$ is diagonalizable? –''

I have a counterexample, where $A$ has positive eigenvalues, but it is not positive definite: $ A= \begin{bmatrix} 7 & -2 & -4 \\ -17 & 40 & -19 \\ -21 & -9 & 31 \end{bmatrix} $. Eigenvalues of this matrix are $1.2253$, $27.4483$, and $49.3263$, but it indefinite because if $x_1 = \begin{bmatrix}-48 & -10& -37\end{bmatrix}$ and $x_2 = \begin{bmatrix}-48 &10 -37\end{bmatrix}$, then we have $π‘₯_1𝐴π‘₯_1^T = -1313$ and $π‘₯_2𝐴π‘₯_2^T = 37647.$


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