Question 1

If A is any matrix, then  is diagonalizable.

Accepted Answer

The answer of If A is any matrix, then ...

Question 2

If A is an  diagonalizable matrix, then there exists a diagonal matrix D and an orthogonal matrix P such that  .

Accepted Answer

The answer of If A is an  diagonalizable matrix,...

Question 3

If Q is an orthogonal matrix, then  .

Accepted Answer

By definition, an orthogonal matrix is a square matrix whose columns and rows are orthonormal vectors. Therefore, its transpose is its inverse, which implies that Q^T*Q = I, where I is the identity matrix.

Question 4

If A is symmetric, and  and  with  , then  is orthogonal to  .

Accepted Answer

Since A is symmetric, its eigenvectors corresponding to distinct eigenvalues are orthogonal. Therefore, if 11edaba5_20a2_6317_a975_5743dffb58ea_TBMC1039_00 and 11edaba5_20a2_6318_a975_eb7c8035abd0_TBMC1039_00 are eigenvectors of A corresponding to distinct eigenvalues, then they are orthogonal. Similarly, if 11edaba5_20a2_6319_a975_4dd483f6be54_TBMC1039_00 is orthogonal to both of them, then it must be orthogonal to their span, which includes 11edaba5_20a2_631a_a975_2d0122acf3d8_TBMC1039_00 and 11edaba5_20a2_631b_a975_172d1fbdaa23_TBMC1039_00. Therefore, 11edaba5_20a2_631a_a975_2d0122acf3d8_TBMC1039_00 is orthogonal to 11edaba5_20a2_631b_a975_172d1fbdaa23_TBMC1039_00.

Question 5

If  is a QR factorization of a matrix A, then  .

Accepted Answer

In QR factorization, we decompose a matrix A as a product of two matrices Q and R, where Q is an orthogonal matrix and R is an upper triangular matrix. The QR factorization of a matrix is not unique, but if 11edaba5_20a2_631c_a975_99c851a0b30d_TBMC1039_00 is the QR factorization of matrix A, then 11edaba5_20a2_631d_a975_a57e0558cc22_TBMC1039_00 is also a valid QR factorization of the same matrix A. The difference between the two factorizations may be due to different choices of sign or scaling factors.

Question 6

Find the singular values for the matrix  .

Accepted Answer

The answer of Find the singular values for the matrix...

Question 7

Find the singular values for the matrix  .

Accepted Answer

The answer of Find the singular values for the matrix...

Question 8

Find a singular value decomposition for the matrix  .

Accepted Answer

The answer of Find a singular value decomposition for the...

Question 9

Find a singular value decomposition for the matrix  .

Accepted Answer

The answer of Find a singular value decomposition for the...

Question 10

Find a singular value decomposition for the matrix  .

Accepted Answer

The answer of Find a singular value decomposition for the...

Question 11

Find a singular value decomposition for the matrix  .

Accepted Answer

The answer of Find a singular value decomposition for the...

Question 12

Express the given matrix  as an outer product expansion  , where  and  are the singular values of  .

Accepted Answer

The answer of Express the given matrix  as an...

Question 13

Express the given matrix  as an outer product expansion  , where   and  are the singular values of  .

Accepted Answer

The answer of Express the given matrix  as an...

Question 14

Determine the numerical rank of a  matrix with singular values  ,  ,  , and  , if  .

Accepted Answer

The answer of Determine the numerical rank of a ...

Question 15

Determine the numerical rank of a  matrix with singular values  ,  ,  ,  , and  , if  .

Accepted Answer

The answer of Determine the numerical rank of a ...

Question 16

Every matrix A has a singular value decomposition.

Accepted Answer

Every matrix A can be decomposed as A=UΣV^T, where U and V are orthogonal matrices, and Σ is a diagonal matrix containing the singular values of A. This is known as the Singular Value Decomposition (SVD).

Question 17

The singular value decomposition of a matrix A is unique.

Accepted Answer

The singular value decomposition (SVD) of a matrix A is not unique because there can be multiple sets of orthogonal matrices that satisfy the decomposition, especially considering sign ambiguities and the ordering of singular values when they are not distinct.

Question 18

If D is a diagonal matrix with diagonal entries  , then the singular values of D are given by  .

Accepted Answer

The singular values of a diagonal matrix are simply the absolute values of its diagonal entries. Therefore, the singular values of D are |11edaba5_20a2_ff94_a975_f74fa41ef432_TBMC1039_00| = 11edaba5_20a2_ff95_a975_0d3b0ea8bee5_TBMC1039_00.

Question 19

If  is a singular value decomposition for a matrix  , then  is an orthogonal diagonalizing matrix for  .

Accepted Answer

By definition, the singular value decomposition of a matrix is given by A = UΣV^T, where U and V are orthogonal matrices and Σ is a diagonal matrix. Therefore, 11edaba5_20a2_ff96_a975_97693f3dd49f_TBMC1039_00 represents the singular values of the original matrix in diagonal form. If we multiply U and V^T, we obtain an orthogonal matrix that diagonalizes the original matrix, and this matrix is represented by 11edaba5_20a2_ff98_a975_efb49f44779b_TBMC1039_00. Therefore, the statement is true.

Question 20

If  is an invertible (square) matrix with singular value decomposition  , then a singular value decomposition for  is given by  .

Accepted Answer

The answer of If  is an invertible (square) matrix...

Quiz 8: Orthogonality

If A is any matrix, then is diagonalizable.

If A is an diagonalizable matrix, then there exists a diagonal matrix D and an orthogonal matrix P such that .

If Q is an orthogonal matrix, then .

If A is symmetric, and and with , then is orthogonal to .

If is a QR factorization of a matrix A, then .

Find the singular values for the matrix .

Find the singular values for the matrix .

Find a singular value decomposition for the matrix .

Find a singular value decomposition for the matrix .

Find a singular value decomposition for the matrix .

Find a singular value decomposition for the matrix .

Express the given matrix as an outer product expansion , where and are the singular values of .

Express the given matrix as an outer product expansion , where and are the singular values of .

Determine the numerical rank of a matrix with singular values , , , and , if .

Determine the numerical rank of a matrix with singular values , , , , and , if .

Every matrix A has a singular value decomposition.

The singular value decomposition of a matrix A is unique.

If D is a diagonal matrix with diagonal entries , then the singular values of D are given by .

If is a singular value decomposition for a matrix , then is an orthogonal diagonalizing matrix for .

If is an invertible (square) matrix with singular value decomposition , then a singular value decomposition for is given by .