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Matrices and Calculus Interview questions and Answers
Table of Content
 What are the characteristics of EingenValue & Eigenvector?
 Which of the following are the steps involved in reducing a quadraticformoftheequationtocanonicalformbyanorthogonaltransformation?
 Which of the following is true about representation of functions?
 What is true about Cayley–Hamilton theorem?

What are the characteristics of EingenValue & Eigenvector?
 a) If the eigenvalue is negative then the direction of the eigenvector will remain the same
 b) If the eigenvalue is negative then the direction of the eigenvector will get reversed
 c) Eigenvector is a zero vector & it is a factor by which eigenvalue changes
 d) In the classical method, first you find the eigenvectors & then the eigenvalues
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 (nonzero) 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.

Which of the following are the steps involved in reducing a quadratic form of the equation to canonical form by an orthogonal transformation?
 a) Matrix form 'A' of quadratic function has to be derived. Post this get the characteristic equation from the matrix derived from the quadratic function
 b) Calculate the eigenvalues & eigenvectors & verify the calculations using the pairwise orthogonal property. Take the eigenvectors and form the modal matrix M. Post that normalizes the modal matrix to get 'N' and then transpose it to 'NT'.
 c) Multiply matrix form of quadratic function & normalized matrix (A*N). Now calculate (NT)*A*N to get the diagonalization
 d) All of the options are true
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.

Which of the following is true about representation of functions?
 a) Functions can be represented in a word description format.
 b) Functions can be represented in a visual format.
 c) Functions can be represented in a tabular format.
 d) All of the options are true
Answer  d) All of the options are true
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"".

What is true about Cayley–Hamilton theorem?
 a) The characteristic equation need not be a square matrix
 b) The characteristic equation should necessarily be a square matrix with only real values (not complex values)
 c) Polynomial expression involving the square matrix will be equal to a zero matrix
 d) Polynomial expression involving the square matrix will not be equal to a zero matrix
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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