Ideal of an inversable triangular matrix

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In summary, an inversable triangular matrix is a square matrix with zero elements above or below the main diagonal, making it easily invertible. The inverse can be found through the Gauss-Jordan elimination method and it has properties of being diagonalizable and having a determinant equal to the product of its diagonal elements. In linear algebra, it is important for solving systems of linear equations and decomposing a matrix. However, a matrix cannot be both triangular and orthogonal at the same time due to contradictory properties.
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karin
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Hello
I need your help please.
I have a block matrix P=[a b ; 0 d], which is inversable.
if f belongs to the ideal of the matrix, how do I prove that
f=[itex]\sum[/itex]g[itex]_{ij}[/itex]x[itex]_{ij}[/itex]
while g[itex]_{ij}[/itex] are regular functions above the space of inversable matrix?
thank you!
Karin
 
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  • #2
I think you should explain the terminology here. What is "the ideal of the matrix", and what are "regular functions above the space of inversable matrix"?
 

What is the definition of an inversable triangular matrix?

An inversable triangular matrix is a square matrix in which all the elements above or below the main diagonal are zero. This means that the matrix can be easily inverted or its inverse can be easily calculated.

How do you find the inverse of a triangular matrix?

The inverse of a triangular matrix can be found by using the Gauss-Jordan elimination method. This involves transforming the original matrix into an identity matrix through a series of elementary row operations, and the resulting matrix will be the inverse of the original triangular matrix.

What properties does an inversable triangular matrix have?

An inversable triangular matrix has the property of being easily invertible, as well as being diagonalizable. It also has a determinant that is the product of its diagonal elements.

What is the significance of an inversable triangular matrix in linear algebra?

Inversable triangular matrices are important in linear algebra because they are useful in solving systems of linear equations. They also play a crucial role in the decomposition of a matrix into its LU factors.

Can a matrix be both triangular and orthogonal at the same time?

No, a matrix cannot be both triangular and orthogonal at the same time. A triangular matrix has zeros in certain positions, while an orthogonal matrix has a special property where its transpose is equal to its inverse. These properties are contradictory, so a matrix cannot possess both at once.

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