Flashcards
MATRIX FUNDAMENTALS
Cards in this deck (38)
- 1what are the eigenvalues of a diagonal matrix?Its diagonal elements
- 2—
- 3Determinant of a diagonal matrix will be?Product of it's diagonal elements
- 4Define the criteria for a UNIT/IDENTITY matrix?A square matrix with all it's diagonal elements as 1 and non diagonal elements as 0
- 5properties of identity matrix(AI) = (IA) = (A)
|A|= 1 - 6does a null matrix has to be a square matrix?ANy matrix with all it's elements as 0 is a null matrix
- 7how can we find inverse for a diagonal matrix?for a diagonal matrix
diag[x,y,z] the inverse is is obtained by replacing each element in the diagonal with its reciprocal i.e. diag[1/x,1/y,1/z] - 8does a diagonal matrix always have inverse as reciprocal of its diagonal entries?as long as none of the diagonal elements is equal to zero. If any of the diagonal elements are equal to zero, the matrix will be less than full rank, and the matrix will not have an inverse.
- 9does all elements above and below diagonal have to be non zero for upper and lower triangular matrix?No, those elements as well as elements of diagonal itself may or may not be zero
- 10what's the product of multiplying two lower triangular matrices?a lower triangular matrix
- 11what will be the result of transpose(Upper triangular)a lower triangular matrix
- 12what does M=LU representa matrix can be expressed as mulpilcation product of a lower and an upper triangular matrix
- 13what's an idempotent Matrix?a matrix for which
A x A = A - 14properties of idempotent matrix?It has to be a square matrix.
An idempotent matrix is always diagonalizable and its eigenvalues are either 0 or 1
The trace of an idempotent matrix — the sum of the elements on its main diagonal — equals the rank of the matrix and thus is always an integer - 15Define a singular matrix and why is it useful?if for a matrix A |A|=0,
then it is singular
a singular matrix has no inverse. - 16criteria for matrix multiplication?numbers of rows of A be equal to the number of Columns in B.
- 17Does A x B =0 imply either A or B is surely a zero matrix?NO, two non zero matrices can give a zero matrix upon multiplication
- 18What's a trace of a matrix? Why is it important?For a square matrix, the trace is the sum of elemrnts lying across the diagonal element.
The trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities), and it is invariant with respect to a change of basis. - 19give property of tracetr{AB} = tr{BA}
given both AB and BA are defined - 20What is the transpose of a matrix?Matrix obtained by interchanging rows and columns of original matrix
- 21Define Transpose(AB)transpose(B) x Transpose(A)
- 22Define Transpose(ABC)transpose(C) x Transpose(B)x Transpose(A)
- 23Properties of a Symmetric Matrixif A, B are symmetric matrix, then
transpose (A) = A
A x Transpose(A) is always symmetric
A+ B, A-B also symmetric
AB , BA may or may not be symmetric - 24Define skew-symmetric matrixif transpose (A)= -A
A being a square matrix, and all diagonal elements 0 - 25for a square matrix A [ (A) - transpose(A) ] / 2 will be askew-symmetric matrix
- 26for any square matrix A=P+Q, define P and QP= [ (A) +transpose(A) ] / 2 symmetric matrix
Q= [ (A) - transpose(A) ] / 2 skew symmetric matrix - 27define an orthogonal matrixtwo vectors are orthogonal, we mean that they are perpendicular,
for a square matrix
A x Transpose(A) = I (identity matrix)
or A= Transpose(A) - 28whats the modulus of the determinant of an orthogonal matrix|A| = plus/minus 1
- 29doea a matrix A having |A| = pus/minus 1 orthogonal?May or may not be
- 30given A and B are orthogonal, comment on the nature of AB and BABoth will be orthogonal as well
- 31define conjugate of a matrix
- 32if conjugate (A) = A then? also if conjugate (A) =-AA is a real matrix
A is purely imaginary - 33define transposed conjugate and it's properties
- 34Condition of hermitian matrix?transposed conjugate (A) = A
- 35condition of a skew hermitian matrix?tranposed conjuagate(A) = -A
- 36Condition for a unitary matrix?tranposed conjugate(A) = inverse(A)
- 37diagonal elements of skew hermitian matrix will be?purely imaginary or zero
- 38—
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- Engineering Flashcards on MATRIX FUNDAMENTALS, created by Roman Rains on 04/08/2021.
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- Roman Rains
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