Orthonormal Matrix Homework: Estimating |A*A'|

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In summary, the conversation discusses the properties of a rectangular matrix with orthonormal columns and the relationship between its transpose and the identity matrix. The speaker is seeking help in finding an upper bound for the norm of the matrix A*A' with respect to the number of orthonormal columns.
  • #1
tom08
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Homework Statement



If A is a rectangular n*m matrix (n>m) , and all the columns of A is orthonormal.

I know that A'*A=I, where A' stands for its transpose.

but A*A'<>I as I've learned from wiki. but is there an estimate for [tex]\|A \cdot A'\| [/tex]?


Homework Equations



http://en.wikipedia.org/wiki/Orthogonal_matrix

in the rectangular matrix section.

The Attempt at a Solution



I have tried to write A in component form to find any hints. but i failed to solve the problem. but when i test A*A' , i always find that norm(A*A')=1, could you help me to explain it? Thank you in advance.
 
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  • #2
tom08 said:

Homework Statement



If A is a rectangular n*m matrix (n>m) , and all the columns of A is orthonormal.

I know that A'*A=I, where A' stands for its transpose.

but A*A'<>I as I've learned from wiki.
What? The Wikipedia site you link to below clearly says, "[itex]Q^TQ= QQ^T= I[/itex]. Alternatively [itex]Q^T= Q^{-1}[/itex]".

but is there an estimate for [tex]\|A \cdot A'\| [/tex]?


Homework Equations



http://en.wikipedia.org/wiki/Orthogonal_matrix

in the rectangular matrix section.

The Attempt at a Solution



I have tried to write A in component form to find any hints. but i failed to solve the problem. but when i test A*A' , i always find that norm(A*A')=1, could you help me to explain it? Thank you in advance.
 
  • #3
no, please look at my wiki link in the last secion, "rectangular matrix"

if Q is not square, but column orthonormal. let Q be an n-by-m matrix, and (m<n),

then Q'*Q=I, but Q*Q'<>I.

so i want to find out an upper bound of ||I-Q*Q'|| w.r.t m, where m is the number of orthonormal columns of Q.
 
  • #4
can someone give me a hand?
 

FAQ: Orthonormal Matrix Homework: Estimating |A*A'|

What is an orthonormal matrix?

An orthonormal matrix is a square matrix whose columns are orthogonal unit vectors. This means that the dot product of any two columns is equal to 0, and the magnitude of each column is equal to 1.

How is an orthonormal matrix used in estimating |A*A'|?

An orthonormal matrix is used to simplify the calculation of |A*A'|. Since the columns of an orthonormal matrix are orthogonal, the dot product term becomes easier to calculate, and the magnitude term becomes 1, resulting in a simpler and more efficient calculation.

What is the significance of |A*A'| in matrix operations?

|A*A'| is also known as the Frobenius norm of matrix A. It is used to measure the overall magnitude of the matrix, and is often used in optimization and data analysis problems.

How is the estimate of |A*A'| derived using an orthonormal matrix?

The estimate of |A*A'| is derived by first decomposing the matrix A into its singular value decomposition (SVD) form. Then, the orthonormal matrices from the SVD are used to simplify the dot product term in the calculation of |A*A'|. The result is an estimate that is close to the actual value of |A*A'|.

Are there any limitations to using an orthonormal matrix in estimating |A*A'|?

While an orthonormal matrix can simplify the calculation of |A*A'|, it is not always the most efficient method. For larger matrices, other techniques such as using the QR decomposition may be more efficient. Additionally, the orthonormal matrix must be carefully chosen to accurately estimate |A*A'|, which may require additional time and effort.

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