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Answer - b) If the eigenvalue is negative then the direction of the eigenvector will get reversed
Eigenvector and eigenvalue are part of linear algebra. Eigenvectors are also called Characteristic vectors. If you perform a linear transformation then the eigenvector of this linear transformation happens to be a vector (non-zero) that changes by some scalar factor. This scalar factor is represented using Lamba and is called an eigenvalue. If the geometrical explanation is to be provided then the eigenvector points in the same direction in which it is transformed by a factor called an eigenvalue. If the eigenvalue is negative then the direction of the eigenvector is reversed. Eigenvector is not rotated in a multidimensional vector space. Eigenvalues & eigenvectors are used in various applications such as geometric transformations, light, microwaves, quantum mechanics, molecular physics, geology, principal component analysis (data science), the vibration of a fork, image processing (eigenfaces), speech recognition systems (eigenvoices), solid mechanics, etc.
Answer - d) All of the options are true
A homogeneous polynomial of the second degree of any number of variables is called a quadratic form. An example is in figure 1. The steps needed to find the canonical form by reducing the quadratic equation are as follows: a. Matrix form 'A' of quadratic function has to be derived. b. Get the characteristic equation from the matrix derived from the quadratic function. c. Calculate the eigenvalues & eigenvectors & verify the calculations using the pairwise orthogonal property. d. Take the eigenvectors and form the modal matrix 'M'. e. Post that normalizes the modal matrix to get 'N' and then transpose it 'NT'. f. Multiply matrix form of quadratic function & normalized matrix (A*N). g. Now calculate (NT)*A*N to get the diagonalization matrix 'D'. Diagonal values will be eigenvalues of 'A' h. Canonical form is written as [y1, y2, y3]*D*[y1, y2, y3]T where T is transposed.
A function is a relationship between 2 variables wherein one variable is dependent on another variable. E.g., y=2x. Functions can be represented in various ways, which might include formulae of graphs. The main 4 ways of representing functions include: a. Algebraic - Function is represented using a mathematical equation. y = f(x) = x^3 is an example of this representation. Here f is a function that maps y to x^3. x^3 is the formula of the function. b. Numerical - Function is represented using a table. Refer to figure 1. c. Visual - Function is represented using a graph. Refer to figure 2. d. Verbal - Function is represented using a textual description. Figure 2 can be described in textual format as ""the lowest value of function is obtained at x = -2 & the highest value of function is obtained at x = 2"".
Answer - c) Polynomial expression involving the square matrix will be equal to a zero matrix
According to Cayley–Hamilton theorem for any given square matrix A, there exists a characteristic polynomial which is given in Figure 1 and instead of scalar lambda if we take matrix A then the resultant characteristic polynomial will always be a zero matrix. Also, this theorem applies to a square matrix with real values as well as complex values.
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