plot matrix with arrows describing it

Let me extend my comment to answer:

• you need to move matrix baseline to their vertical center
• for this is sufficient to add option baseline to tikzpicture:

documentclass[12pt]{article}
usepackage{amsmath}
usepackage{amscd}
usepackage{tikz}
usetikzlibrary{arrows.meta, matrix, quotes, positioning}
begin{document}
        begin{equation}label{key}
            left[bigl( t_{A}vert E_{lambda_{i}}(A)bigr)middle| X^{i}, X^{i} right] =
            begin{tikzpicture}[baseline, %<------
            LA/.style = {Straight Barb-Straight Barb,
                         shorten <=1pt, shorten >=1pt,
                         draw=blue}
                                ]
                matrix (m) [matrix of math nodes,
                            left  delimiter={[},
                            right delimiter={]}
                ]
                {
                    lambda_{i}&&0&
                    &lambda_{i} &&
                    0 && ddots &
                    &&& lambda_{i}
                };
                draw[LA] (m-3-1.south west |- m.south) to [" $d_{i}$ " '] (m-4-4.south east |- m.south);
                coordinate[right=5mm of m.east] (e);
                draw[LA] (m-1-3.north east -| e) to [" $d_{i}$ "] (m-4-4.south east -| e);
            end{tikzpicture},  i=1,dotsc,k
        end{equation}
end{document}

Addendum:

You may liked version written with help of the nicematrix package:

documentclass[12pt]{article}
usepackage{nicematrix}
usepackage{tikz}

begin{document}
    begin{equation}
NiceMatrixOptions
{code-for-last-col=color{blue},
 code-for-last-row=color{blue}}
left[
bigl( t_{A}vert E_{lambda_{i}}(A)bigr)middle| X^{i}, X^{i} 
right] = 
begin{bNiceMatrix}[last-row,last-col]
lambda_{i} &               & Block{2-2}<huge>{0} &          
            & lambda_{i}   &        &              & quadVdots[line-style={solid,<->}]^{d_i}     
Block{2-2}<huge>{0} 
            &               & ddots &              &   
            &               &        & lambda_{i}  &   
& Ldots[line-style={solid,<->}]_{d_i}              &   
end{bNiceMatrix} ,  i=1,dotsc,k
    end{equation}
end{document}

For final result you need to compile this MWE at least twice.

documentclass{article}
usepackage{tikz}
usepackage{etoolbox}
letbbordermatrixbordermatrix
patchcmd{bbordermatrix}{8.75}{4.75}{}{}
patchcmd{bbordermatrix}{left(}{left[}{}{}
patchcmd{bbordermatrix}{right)}{right]}{}{}
newcommand{tikzmark}[1]{tikz[overlay, remember picture] coordinate (#1);}
begin{document}
[
    left[ left. left( t_{A}vert E_{lambda_{i}}(A)right)right| X^{i}, X^{i} right] =quad bbordermatrix{~  &tikzmark{harrowleft}  &  &  & 
        tikzmark{harrowright}cr
        tikzmark{varrowtop} & lambda_{i }&  & 0 & cr
        && lambda_{i} &&  cr
        & 0 && ddots &cr
        tikzmark{varrowbottom} &  &  & &  lambda_{i}  cr
    }
    ,quad i=1,dots, k.
]
tikz[overlay,remember picture] {
    draw[<->] ([yshift=2ex]harrowleft) -- ([yshift=2ex]harrowright)
    node[midway,above] {scriptsize $ d_{i} $};
    draw[<->] ([yshift=1.5ex,xshift=-2ex]varrowtop) -- ([xshift=-2ex]varrowbottom)
    node[near end,left] {scriptsize $d_{i}$};
}
end{document}