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I'm having trouble with some maths regarding the expression of the matrix quadratic form (i.e. $x^TAx$) and the proof that, where the eigenvalues $\lambda_1,\lambda_2,...,\lambda_n$ are all positive, the quadratic form is positive definite.

My understanding is that the definition of positive-definiteness is when $x^TAx>0$ for all x where at least one element of $x \neq 0$.

My textbook produces the following proof, but I don't understand the last line:

Where $s_i$ = a normalized eigenvector of A, $\lambda_i$ = the corresponding eigenvalue, and $C = [s_1|s_2|...|s_n]$, $C^{-1}AC=D=$ the "diagonalization" of A.

Since the eigenvectors of A are orthonormal, $C^{-1} = C^T$.

Suppose we define a transformation $x=CX$. Then the equation becomes: $x^TAx = (CX)^{T}ACX = X^TDX = \lambda_1X_1^2 + \lambda_2X_2^2+...+\lambda_nX_n^2$.

It follows from this that a quadratic form is positive-definite if and only if all its eigenvalues are positive.

So, in summary, I don't understand why the following derivation true:

$x^TAx = \lambda_1X_1^2 + \lambda_2X_2^2+...+\lambda_nX_n^2$ therefore, for $x^TAx > 0$ for all x where at least one element of $x_i \neq 0$ to be true, $\lambda_1, \lambda_2, ..., \lambda_n > 0$.

Can someone please help with the derivation of the last step?a

Thanks in advance for your help.

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I think your definition of positive definiteness may be the source of your confusion. The correct definition is that $A$ is positive definite if $x^TAx>0$ for all vectors $x$ other than the zero vector. –  Will Orrick Apr 30 '12 at 2:20
    
@WillOrrick that is what I mean by at least one element of x is not zero. To restate: for all x except x = [0;0]. Thanks for you comment though. –  JonaGik Apr 30 '12 at 2:25
    
"...therefore, for $x^TAx\gt0$ with at least one element of $x_i\ne0$ to be true, $λ_1$, $λ_2$,...,$λ_n\gt0$." --- I still think you really need to change this to "...for $x^TAx\gt0$ to be true for all $x$ having at least one element $x_i\ne0$..." The "for all $x$" allows you to choose an $x$ that will help you reach the conclusion. Since $x^TAx\gt0$ has to be true for any $x$, you can choose $x$ in such a way that only one of the eigenvalues contributes. Then you can conclude that that eigenvalue must be positive. Repeat for every eigenvalue. –  Will Orrick Apr 30 '12 at 4:10
    
Thanks for adding that. I think you also need to correct it in the second gray box. This may help clarify the logic of the proof. –  Will Orrick Apr 30 '12 at 5:29
    
@WillOrrick no problem and fixed in the second gray box too :) –  JonaGik May 1 '12 at 1:08
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3 Answers

For the "if" part: if all of the $\lambda_j$ are positive then $\lambda_1X_1^2+\ldots+\lambda_nX_n^2$ is positive for all non-zero vectors $X$ and hence for all non-zero vectors $x$.

For the "only if" part: Suppose that $\lambda_1X_1^2+\ldots+\lambda_nX_n^2$ is positive for all non-zero vectors $x$. Then in particular, it is positive for the vector $x$ corresponding to the $X$ with $X_j=1$ and $X_k=0$ for all $k\ne j$. Therefore $\lambda_j$ is positive.

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Thanks for your response. I don't understand how the latter is an "only if" proof - it seems to only prove that the expression is positive in the particular (not general) case where X is any vector with a single element equal to one and the others equal to zero. Consider the case where $X = [1, 1, 0]$ (which appears to be legal). Now we have $\lambda_1 + \lambda_2 > 0$ which isn't constrained to all $\lambda_i > 0$. Presumably this is a valid counter-example? –  JonaGik Apr 30 '12 at 5:43
    
The eigenvalues are fixed numbers, depending only on $A$. They are what they are. One thing that we know about them is that $\lambda_1X_1^2+\ldots+\lambda_nX_n^2$ is positive for all $X$ except $X=0$. What conclusions about the values of $\lambda_j$ can we draw from this fact? Both your conclusion, $\lambda_1+\lambda_2\gt0$, and my conclusion, $\lambda_1\gt0$, are valid inferences. Mine just happens to be a stronger conclusion. For every non-zero choice of $X$ we get a conclusion. All of these conclusions are true statements about the $\lambda_j$. –  Will Orrick Apr 30 '12 at 7:23
    
My understanding is that we want to classify A (as positive-definite, negative-definite, etc) on the basis of its eigenvalues. In that case, we don't know that $\lambda_1X_1^2+\ldots+\lambda_nX_n^2$ is positive. We wish to determine this based on its eigenvalues. What you have given is a case where all the eigenvalues are positive and the matrix is positive definite, but not a general, proven rule for this. –  JonaGik Apr 30 '12 at 9:12
    
@JonaGik The goal stated in your original post was to prove that a quadratic form is positive-definite if and only if all its eigenvalues are positive. In a previous comment, you questioned the proof of the "only if" part: a quadratic form is positive-definite only if all its eigenvalues are positive. This part is logically equivalent to the statement that if a quadratic form is positive-definite, then all of its eigenvalues are positive. In order to prove this, we take a positive-definite form. By definition of positive-definite, $x^TAx\gt0$ for all non-zero $x$. The manipulations... –  Will Orrick Apr 30 '12 at 12:40
    
(continued) ...in the proof quoted in your post establish that $\lambda_1X_1^2+\ldots+\lambda_nX_n^2\gt0$ for all non-zero $X$. This is not an assumption; it follows from the definition of positive-definiteness. We may then choose particular values for the $X_j$ in order to draw inferences about the $\lambda_j$. –  Will Orrick Apr 30 '12 at 12:44
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I misread the question. @WillOrrick's comment is relevant, if $A$ is positive definite, then you must have $x^T Ax > 0$ for any $x \neq 0$ (which is equivalent to at least one $x_i \neq 0$).

To illustrate why the 'for any' is important, consider the matrix $B = \left( \begin{array}{cc} 1 & 0 \\ 0 & -1 \end{array} \right)$. Then $e_1^T B e_1 > 0$, but $B$ is not positive definite. The 'for any $x \neq 0$' is important.

Since you have $x^TAx = \sum_{i=1}^n \lambda_i X_i^2$ (with $x = CX$), we may choose $x=C e_k$, where $e_k$ is the $k$th basis vector (ie, $x$ is an eigenvector corresponding to $\lambda_k$). It follows that for this choice of $x$, we have $x^TAx = \sum_{i=1}^n \lambda_i X_i = \lambda_k X_k = \lambda_k$. Since $A$ is positive definite, it follows that $\lambda_k>0$. Since $k$ was arbitrary, we have that all eigenvalues are strictly positive.

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Thanks very much for your response. This seems like its definitely on the right track. For $\sum_{i-1}^n X_i^2 = \underline \lambda \sum_{i-1}^n x_i^2$, I'm having trouble proving that to myself. Do you think you could prove it? Thanks again! –  JonaGik Apr 30 '12 at 3:01
    
As far as I can tell, your proof assumes that A is a diagonal matrix so that the eigenvectors are [1, 0, ..., 0], [0, 1, 0, ..., 0], ..., [0, ..., 0, 1] and thus $x^TAx = \sum_{i=1}^n \lambda_i X_i = \lambda_k X_k = \lambda_k$ is true. It is possible for more than one $\lambda_iX_i^2$ combination to be non-zero in the general case. Is this right? If so, this doesn't produce a general proof. Also, does this proof prove only that if all eigenvectors are positive, the matrix is positive-definite, not if and only if all eigenvectors are positive the matrix is positive-definite? –  JonaGik Apr 30 '12 at 4:01
    
Since $C$ is a basis of orthonormal eigenvectors, the matrix $C^T A C $ is diagonal. Note the distinction between $x$ and $X$ in the proof. We have $x^T Ax = (CX)^T A (CX) = X^T (C^T A C) X$. –  copper.hat Apr 30 '12 at 4:15
    
Your question asked for elaboration of the last step, not an 'iff' proof? The elaboration shows that if $A$ is positive definite, then you must have $\lambda_i > 0$. –  copper.hat Apr 30 '12 at 4:16
    
Are you saying that $e_k$ is some arbitrary eigenvector of $D = C^TAC$? Because then I can see this working. However, overall, the proof seems to still only prove that if (but not iff) all the eigenvectors are positive, the matrix is positive-definite. Is this right? –  JonaGik Apr 30 '12 at 5:11
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The "only if" part (if $A$ is pos.def, then all eigenvalues are positive) can be proved more easily by absurd. Suppose that $x^t A x>0$ for all $x\ne 0$, but we have some $\lambda_i <0 $. The, calling $p_i$ ($\ne 0$) the associated eigenvector, we have $A p_i = \lambda_i p_i$ and multipliyin by $p_i^t$ we get $ p_i^t A p_i = \lambda_i |p_i|^2 <0 $ which contradicts the initial assumption.

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Unless I'm missing something, this only proves that this is inconsistent when x is the negative eigenvalue's eigenvector - it isn't a proof for the general case. –  JonaGik May 8 '12 at 9:31
    
No, it's general. Given the assumption ("Suppose that... for all $x$") we find one $x$ that contradicts that assumption. That's enough to prove that the assumption is false. –  leonbloy May 8 '12 at 13:44
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