TIFR CSE 2010 | Part A | Question: 5

Question

$A$ is symmetric positive definite matrix ( i.e., $x^{T} Ax > 0$ for all non zero $x$). Which of the following statements is false?

• A. At least one element is positive.
• B. All eigen values are positive real. 
• C. Sum of the diagonal elements is positive.
• D. det (A) is positive.
• E. None of the above.

Comments

Well, I think (e) is the answer. Wiki

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how (e) is true?

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question is asking for false and since all the options are true so e is the answer.i.e. none of the above options are false

Answer 1

$x^TAx>0$ means that matrix is positive definite. And properties of positive definite matrix (relevant to question) is :

• All its eigen values are positive.

Let's consider a $3*3$ matrix for simplicity.

1. Trace(A) = sum of eigen values = positive
2. Det(A) = product of eigen values = positive

Take any matrix and try to anlayse the validity of these points.

So, (E) is the answer here. 

Comments

@goxul 

what is significance of this  : 

x⊤Ax > 0

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$x^tAx=x^t(\lambda x)>0$

$\Rightarrow\lambda xx^t>0$

$\Rightarrow \lambda [x_1^2+x_2^2+\ldots]_{1\times1}>0$

$\Rightarrow\lambda >0$

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A ${\displaystyle n\times n}$ symmetric real matrix ${\displaystyle M}​$​​​​​​ is said to be positive definite if ${\displaystyle x^{\textsf {T}}Mx>0}​​$​​​​​ for all non-zero ${\displaystyle x}$ in ${\displaystyle \mathbb {R} ^{n}}​​​​​​$​

Formally,

${\displaystyle M{\text{ positive definite}}\quad \iff \quad x^{\textsf {T}}Mx>0{\text{ for all }}x\in \mathbb {R} ^{n}\setminus \mathbf {0} }$

• A positive definite matrix is a symmetric matrix $\mathbf{A}$ for which all eigenvalues are positive.

References: 1  2  3 

Answer 2

ALL eigenvalues will be positive as X^TAX>-0

As eigen values ppositive so the trace of matrix i.e sum of diagonal elements are positive

Det(a) also positive as all eigen values are positive

Only Option (A) may or may not always follow

A is the answer here

Comments

How?