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Let X be a column vector of dimension n > 1 with at least one non-zero entry. The number of non-zero eigenvalues of the matrix M = XXT is
  • a)
    0
  • b)
    n
  • c)
    1
  • d)
    n - 1
Correct answer is option 'C'. Can you explain this answer?
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Most Upvoted Answer
Let X be a column vector of dimension n > 1 with at least one non-z...
Concept:
Eigenvalues are the special set of scalar values that is associated with the set of linear equations most probably in the matrix equations
Explanation:

Here, X is an n × 1 column vector with the entry in the ith row equal to a.
XT is a row vector having an entry in the ith column equal to a. Then, XXT is an n × 1 matrix having the entry in the ith row and ith column equal to a2 .

Since this matrix is diagonal, its eigenvalues are a2, 0, 0.......0. Hence, the number of non-zero eigenvalues of the matrix XXT is 1.
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Let X be a column vector of dimension n > 1 with at least one non-z...
Let X be a column vector of dimension n. In matrix notation, X can be represented as:

X = [x1, x2, x3, ..., xn]T

where xi represents the i-th element of X. The superscript T denotes the transpose operation, which converts a row vector into a column vector.

The dimension of X indicates the number of elements it contains. Therefore, the column vector X has n elements.
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Direction: Read the following passage carefully and answer the question. If India has to develop her internal strengths, the nation has to focus on the technological imperatives, keeping in mind three dynamic dimensions: the people, the overall economy and the strategic interests. These technological imperatives also take into account a ‘fourth’ dimension, time, an offshoot of modern-day dynamism in business, trade, and technology that leads to continually shifting targets. We believe that technological strengths are especially crucial in dealing with this fourth dimension underlying continuous change in the aspirations of the people, the economy in the global context, and the strategic interests. The progress of technology lies at the heart of human history. Technological strengths are the key to creating more productive employment in an increasingly competitive market place and to continually upgrade human skills. Without a pervasive use of technologies, we cannot achieve the overall development of our people in the years to come. The direct linkages of technology to the nation’s strategic strengths are becoming more and more clear, especially since the 1990s. India’s own strength in a number of core areas still puts it in a position of reasonable strength in the geopolitical context. Any nation aspiring to become a developed one needs to have strengths in various strategic technologies and also the ability to continually upgrade them through its own creative strengths. For people-oriented actions as well, whether for the creation of large scale productive employment or for ensuring nutritional and health security for people or for better living conditions, technology is the only vital input. The absence of greater technological impetus could lead to lower productivity and wastage of precious natural resources. Activities with low productivity or low-value addition, in the final analysis, hurt the poorest most. The technological imperatives to lift our people to a new life, and to life, they are entitled to be important. India, aspiring to become a major economic power in terms of trade and increase in GDP, cannot succeed on the strength of turnkey projects designed and built abroad or only through large-scale imports of plant machinery, equipment and know-how. Even while being alive to the short-term realities, medium and long-term strategies to develop core technological strengths within our industry are vital for envisioning a developed India. Q. More productive employment demands

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Let X be a column vector of dimension n > 1 with at least one non-zero entry. The number of non-zero eigenvalues of the matrix M = XXTisa)0b)nc)1d)n - 1Correct answer is option 'C'. Can you explain this answer?
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